Physical Sciences - Grade 10 CAPS

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Chapter 1: Algebraic expressions

1.1 Introduction

Content covered in this chapter includes understanding how numbers are classified as rational or irrational, estimating surds, rounding off, factorisation and simplification.

This chapter provides a lot of core skills that learners will apply to the rest of mathematics. Ensure that learners are sufficiently proficient in the skills covered in this chapter.

Rounding real numbers is an important skill that learners will use often. Ensure that learners are completely comfortable with this skill.

Factorisation forms the groundwork for solving equations. Learners should be comfortable factorising trinomials and binomials.

Factorisation should include types covered in grade 9 as well as trinomials, grouping in pairs and sum and difference of two cubes.

Over human history, all peoples and cultures have contributed to the field of Mathematics. Topics like algebra may seem obvious now, but for many centuries mathematicians had to make do without it. Over the next three grades, you will explore more advanced and abstract mathematics. It may not be obvious how this applies to everyday life, but the truth is, mathematics is required for nearly everything you will do one day. Enjoy your mathematical journey. Remember, there is no such thing as a “maths person” or “not a maths person”. We can all do mathematics, it just takes practice.

Figure 1.1: Some examples of early tally sticks. These were used to help people count things such as the number of days between events or the number of livestock they had.

In this chapter, we will begin by revising the real number system and then learn about estimating surds and rounding real numbers. We will also be expanding on prior knowledge of factorisation and delve into more complex calculations involving binomial and trinomial expressions.

1.2 The real number system

We use the following definitions:

: natural numbers are

: whole numbers are

: integers are

The following video shows an example of determining which of the above sets of numbers a particular number is in.

Video: 2DBH

Not all numbers are real numbers. The square root of a negative number is called a non-real or imaginary number. For example , and are all non-real numbers.

1.3 Rational and irrational numbers

Rational number

A rational number () is any number which can be written as:

where and are integers and .

The following numbers are all rational numbers:

We see that all numerators and all denominators are integers.

This means that all integers are rational numbers, because they can be written with a denominator of .

Irrational numbers

Irrational numbers () are numbers that cannot be written as a fraction with the numerator and denominator as integers.

Examples of irrational numbers:

These are not rational numbers, because either the numerator or the denominator is not an integer.

Decimal numbers

All integers and fractions with integer numerators and non-zero integer denominators are rational numbers. Remember that when the denominator of a fraction is zero then the fraction is undefined.

You can write any rational number as a decimal number but not all decimal numbers are rational numbers. These types of decimal numbers are rational numbers:

Decimal numbers that end (or terminate). For example, the fraction can be written as .

Decimal numbers that have a repeating single digit. For example, the fraction can be written as or . The dot and bar notations are equivalent and both represent recurring 's, i.e. .

Decimal numbers that have a recurring pattern of multiple digits. For example, the fraction can also be written as . The bar represents a recurring pattern of 's and 's, i.e. .

You may see a full stop instead of a comma used to indicate a decimal number. So the number can also be written as 0.4

Notation: You can use a dot or a bar over the repeated digits to indicate that the decimal is a recurring decimal. If the bar covers more than one digit, then all numbers beneath the bar are recurring.

If you are asked to identify whether a number is rational or irrational, first write the number in decimal form. If the number terminates then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational.

When you write irrational numbers in decimal form, you may continue writing them for many, many decimal places. However, this is not convenient and it is often necessary to round off.

Rounding off an irrational number makes the number a rational number that approximates the irrational number.

Worked example 1: Rational and irrational numbers

Which of the following are not rational numbers?

Irrational, decimal does not terminate and has no repeated pattern.

Rational, decimal terminates.

Irrational, decimal does not terminate and has no repeated pattern.

Rational, all integers are rational.

Rational, decimal has repeated pattern.

Rational, decimal has repeated pattern.

Converting terminating decimals into rational numbers

A decimal number has an integer part and a fractional part. For example, has an integer part of and a fractional part of because .

Each digit after the decimal point is a fraction with a denominator in increasing powers of .

For example:

is

is

is

This means that

The following two videos explain how to convert decimals into rational numbers.

Part 1

Video: 2DBJ

Part 2

Video: 2DBK

Converting recurring decimals into rational numbers

When the decimal is a recurring decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction.

Worked example 2: Converting decimal numbers to fractions

Write in the form (where and are integers).

Define an equation

Multiply by on both sides

Subtract the first equation from the second equation

Simplify

Worked example 3: Converting decimal numbers to fractions

Write as a rational fraction.

Define an equation

Multiply by on both sides

Subtract the first equation from the second equation

Simplify

In the first example, the decimal was multiplied by and in the second example, the decimal was multiplied by . This is because there was only one digit recurring (i.e. ) in the first example, while there were three digits recurring (i.e. ) in the second example.

In general, if you have one digit recurring, then multiply by . If you have two digits recurring, then multiply by . If you have three digits recurring, then multiply by and so on.

Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits and they do not terminate.

Exercise 1.1

The figure here shows the Venn diagram for the special sets and .

Where does the number belong in the diagram?

First simplify the fraction:

is an integer, so it falls into the set .

In the following list, there are two false statements and one true statement. Which of the statements is true?

Every integer is a natural number.

Every natural number is a whole number.

There are no decimals in the whole numbers.

Consider each option carefully:

There are integers which do not fall into the natural numbers (all negative numbers), so this is false.

The natural numbers are and whole numbers are (the circle is inside ) so if a number is a natural number it must be a whole number. This is true.

Whole numbers only go up in steps of 1, so there cannot be any decimal numbers in the whole numbers, making this false.

So only (ii) is true.

The figure here shows the Venn diagram for the special sets and .

Where does the number belong in the diagram?

is in its simplest form, therefore it is not in , or . It is in the space between the rectangle and .

In the following list, there are two false statements and one true statement. Which of the statements is true?

Every integer is a natural number.

Every whole number is an integer.

There are no decimals in the whole numbers.

Consider each option carefully:

There are integers which do not fall into the natural numbers (all negative numbers), so this is false.

The integers are and the whole numbers are (the circle is inside ) so if a number is an integer it must be a whole number. This is true.

Whole numbers only go up in steps of 1, so there cannot be any decimal numbers in the whole numbers, making this false.

So only (ii) is true.

State whether the following numbers are real, non-real or undefined.

has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real.

has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real.

has a minus sign under the square root so it is non-real.

has division by zero so it is undefined.

has a negative number under the square root so it is non-real.

has no minus under the square root (the minus is outside the root), is not divided by zero, so it is real.

State whether the following numbers are rational or irrational. If the number is rational, state whether it is a natural number, whole number or an integer.

is rational. A fraction of integers is a rational number.

is irrational. It cannot be simplified to a fraction of integers.

is rational, an integer, a whole number and a natural number. An integer is a rational number.

is irrational. It cannot be simplified to a fraction of integers.

is irrational. It cannot be simplified to a fraction of integers.

is irrational. It cannot be simplified to a fraction of integers.

is rational, an integer, a whole number and a natural number. It can be written as an integer.

is irrational. is rational (it is an integer). Any rational number added to any irrational number is irrational.

Therefore is irrational.

is irrational. is rational (it is a terminating decimal). Any rational number added to any irrational number is irrational.

Therefore is irrational.

If is an integer, is an integer and is irrational, which of the following are rational numbers?

is rational.

Since is an integer, is rational.

Since is an integer, is rational.

Note that cannot be as that makes the fraction undefined.

Since is irrational, is irrational.

For each of the following values of state whether is rational or irrational.

is rational.

is rational.

is irrational.

is rational.

Consider the following list of numbers:

Which of the numbers are:

natural numbers

Check which of the numbers are in the set . Therefore and are natural numbers.

irrational numbers

Remember that rational numbers can be written as where and are integers. Also remember that rational numbers include terminating decimal numbers. Therefore are all irrational.

non-real numbers

Any number that is a square root of a negative number is non-real. Therefore only is non-real.

rational numbers

Remember that rational numbers can be written as where and are integers. Also remember that rational numbers include terminating decimal numbers. Therefore are all rational numbers.

integers

Check which of the numbers are in the set . Therefore are integers.

undefined

Any fraction divided by is undefined. Therefore only is undefined.

For each of the following numbers write the next three digits and state whether the number is rational or irrational.

Since there is a dot over the 5 we know that the 5 repeats. The next three digits are:

Rational, there is a repeating pattern of digits.

The number does not terminate (this is shown by the ). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

Irrational, there is no repeating pattern.

The number does not terminate (this is shown by the ). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

Irrational, there is no repeating pattern.

The number does not terminate (this is shown by the ). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

Irrational, there is no repeating pattern.

Since there is a dot over both the 5 and the 4 we know that the pattern 54 repeats. The next three digits are:

Rational, there is a repeating pattern.

Write the following as fractions:

Write the following using the recurring decimal notation:

We see that only the digit is repeated and so we can write this as: .

There is a repeating pattern of and so we can write this number as:

There is a repeating pattern of and so we can write this number as:

The pattern 4145 repeats and so we can write this number as: .

Write the following in decimal form, using the recurring decimal notation:

Write the following decimals in fractional form:

1.4 Rounding off

Rounding off a decimal number to a given number of decimal places is the quickest way to approximate a number. For example, if you wanted to round off to three decimal places, you would:

count three places after the decimal and place a between the third and fourth numbers;

round up the third digit if the fourth digit is greater than or equal to ;

leave the third digit unchanged if the fourth digit is less than ;

if the third digit is and needs to be rounded up, then the becomes a and the second digit is rounded up.

So, since the first digit after the is a we must round up the digit in the third decimal place to a and the final answer of rounded to three decimal places is .

The following video explains how to round off.

Video: 2DD8

Worked example 4: Rounding off

Round off the following numbers to the indicated number of decimal places:

to decimal places.

to decimal places.

to decimal places.

to decimal places.

Mark off the required number of decimal places

If the number is not a decimal you first need to write the number as a decimal.

Check the next digit to see if you must round up or round down

The last digit of must be rounded down.

The last digit of must be rounded up.

The last digit of must be rounded up.

The last digit of must be rounded up.

Since this is a we replace it with a and round up the second last digit.

Write the final answer

rounded to decimal places.

rounded to decimal places.

rounded to decimal places.

Exercise 1.2

Round off the following to decimal places:

Mark off the required number of decimal places: . The next digit is a and so we round down: .

Mark off the required number of decimal places: . The next digit is a and so we round up: .

Mark off the required number of decimal places: . The next digit is a and so we round up: .

Mark off the required number of decimal places: . The next digit is a and so we round up: .

Mark off the required number of decimal places: . The next digit is a and so we round up: .

Mark off the required number of decimal places: . The next digit is a and so we round up: .

Round off each of the following to the indicated number of decimal places:

to decimal places.

to decimal places.

to decimal places.

to decimal places.

We first write the fraction as a decimal and then we can round off.

to decimal places.

We first write the fraction as a decimal and then we can round off.

to decimal places.

We first write the fraction as a decimal and then we can round off.

Study the diagram below

Calculate the area of to decimal places.

is a square and so the area is just the length squared.

Calculate the area of to decimal places.

is a right-angled triangle and so we have the perpendicular height. The area is:

Using you answers in (a) and (b) calculate the area of .

The area of is the sum of the areas of and .

Without rounding off, what is the area of ?

Given ; ; ; .

Calculate correct to decimal places.

Using you answer from (a), calculate in .

Calculate without rounding off your answer in (a), compare this answer with your answer in (b).

There is a difference between the answer in (b) and the one calculated without rounding until the final step.

If it takes person to carry boxes, how many people are needed to carry boxes?

Each person can carry 3 boxes. So we need to divide 31 by 3 to find out how many people are needed to carry 31 boxes.

Therefore people are needed to carry boxes. We cannot have of a person so we round up to the nearest whole number.

If tickets cost , how much does one ticket cost?

Since 7 tickets cost , 1 ticket must cost divided by 7.

Therefore one ticket costs . Money should be rounded off to decimal places.

1.5 Estimating surds

If the root of a number cannot be simplified to a rational number, we call it a surd. For example, and are surds, but is not a surd because it can be simplified to the rational number .

In this chapter we will look at surds of the form where is any positive number, for example, or . It is very common for to be so we usually do not write . Instead we write the surd as just

It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to estimate where a surd like is on the number line. From a calculator we know that is equal to . It is easy to see that is above and below . But to see this for other surds like , without using a calculator you must first understand the following:

If and are positive whole numbers, and , then

A perfect square is the number obtained when an integer is squared. For example, is a perfect square since .

Similarly, a perfect cube is a number which is the cube of an integer. For example, is a perfect cube, because .

Consider the surd . It lies somewhere between and because and and is between and .

The following video explains how to estimate a surd.

Video: 2DDV

Worked example 5: Estimating surds

Find two consecutive integers such that lies between them. (Remember that consecutive integers are two integers that follow one another on the number line, for example, and or and .)

Use perfect squares to estimate the lower integer

. Therefore .

Use perfect squares to estimate the upper integer

. Therefore .

Write the final answer

Worked example 6: Estimating surds

Find two consecutive integers such that lies between them.

Use perfect cubes to estimate the lower integer

, therefore .

Use perfect cubes to estimate the upper integer

, therefore .

Write the answer

Check the answer by cubing all terms in the inequality and then simplify

. This is true, so lies between and .

Exercise 1.3

Determine between which two consecutive integers the following numbers lie, without using a calculator:

Estimate the following surds to the nearest decimal place, without using a calculator.

Since , must lie between 3 and 4. But we note that 10 is closer to 9 than to 16 and so will be closer to 3 than to 4.

or are suitable estimates.

Since , must lie between 9 and 10. But we note that 82 is closer to 81 than to 100 and so will be closer to 9 than to 10.

is a suitable estimate.

Since , must lie between 3 and 4. But we note that 15 is closer to 16 than to 9 and so will be closer to 4 than to 3.

is a suitable estimate.

Since , must lie between 9 and 10. But we note that 90 is about halfway between 81 and 100, so will be halfway between 3 and 4.

is a suitable estimate.

Consider the following list of numbers:

Without using a calculator, rank all the numbers in ascending order.

Remember that negative numbers are smaller than positive numbers. It may also be helpful to write the fractions as decimals to help you estimate the number. For the surds you can estimate between which two numbers the surd lies and use that to help you rank these numbers.

lies between 4 and 5

lies between and

lies between 7 and 8

Also note that .

Therefore we get the following order:

1.6 Products

Mathematical expressions are just like sentences and their parts have special names. You should be familiar with the following words used to describe the parts of mathematical expressions.

Name

Examples

term

expression

coefficient

exponent

base

constant

variable

equation

Multiplying a monomial and a binomial

A monomial is an expression with one term, for example, or . A binomial is an expression with two terms, for example, or .

Worked example 7: Simplifying brackets

Simplify:

Multiplying two binomials

Here we multiply (or expand) two linear binomials:

Worked example 8: Multiplying two binomials

Find the product:

The product of two identical binomials is known as the square of the binomial and is written as:

If the two terms are of the form and then their product is:

This product yields the difference of two squares.

Multiplying a binomial and a trinomial

A trinomial is an expression with three terms, for example, . Now we can learn how to multiply a binomial and a trinomial.

To find the product of a binomial and a trinomial, multiply out the brackets:

This video shows some examples of multiplying a binomial and a trinomial.

Video: 2DFF

Worked example 9: Multiplying a binomial and a trinomial

Find the product:

Expand the bracket

Simplify

Exercise 1.4

Expand the following products:

Expand the following products:

Expand the following products:

What is the value of , in

From the constant term we see that . We can check the term: .

What is the value of , in

From the constant term we see that , therefore . We can check the term: .

In :

For which of these values of will be positive?

The term is so for to be positive . Therefore .

For which of these values of will be positive?

The constant term is so for to be positive . Therefore or .

For what real values of will be positive?

From the previous question we see that will make positive.

For what real values of will be positive?

From earlier we see that will make positive.

Answer the following:

Expand .

Given that , determine the value of without solving for .

Now we note that the above expression can also be written as . Since we get:

Answer the following:

Expand:

Given that , determine the value of without solving for .

Given that , determine the value of without solving for .

We note that:

Next we note that if we add 4 to we get . Therefore:

Answer the following:

Expand:

Given that , determine the value of without solving for .

Answer the following:

Expand:

Expand:

Given that , determine the value of without solving for .

1.7 Factorisation

Factorisation is the opposite process of expanding brackets. For example, expanding brackets would require to be written as . Factorisation would be to start with and end up with .

The two expressions and are equivalent; they have the same value for all values of .

In previous grades, we factorised by taking out a common factor and using difference of squares.

Common factors

Factorising based on common factors relies on there being factors common to all the terms.

For example, can be factorised as follows:

And can be factorised as follows:

The following video shows an example of factorising by taking out a common factor.

Video: 2DHZ

Worked example 10: Factorising using a switch around in brackets

Factorise:

Use a “switch around” strategy to find the common factor.

Notice that

Exercise 1.5

Factorise:

Difference of two squares

We have seen that can be expanded to .

Therefore can be factorised as .

For example, can be written as which is a difference of two squares. Therefore, the factors of are and .

To spot a difference of two squares, look for expressions:

consisting of two terms;

with terms that have different signs (one positive, one negative);

with each term a perfect square.

For example: ; ; .

The following video explains factorising the difference of two squares.

Video: 2DJK

Worked example 11: The difference of two squares

Factorise: .

Take out the common factor

Factorise the difference of two squares

Exercise 1.6

Factorise:

Note that

Note that

Factorising by grouping in pairs

The taking out of common factors is the starting point in all factorisation problems. We know that the factors of are and . Similarly, the factors of are and . Therefore, if we have an expression:

there is no common factor to all four terms, but we can factorise as follows:

We can see that there is another common factor . Therefore, we can write:

We get this by taking out the and seeing what is left over. We have from the first group and from the second group. This is called factorising by grouping.

Worked example 12: Factorising by grouping in pairs

Find the factors of .

There are no factors common to all terms

Group terms with common factors together

is a common factor of the first two terms and is a common factor of the second two terms. We see that the ratio of the coefficients is the same as .

Take out the common factor

OR

Group terms with common factors together

is a common factor of the first and third terms and is a common factor of the second and fourth terms .

Rearrange the equation with grouped terms together

Take out the common factor

Write the final answer

The factors of are and .

Exercise 1.7

Factorise the following:

Factorising a quadratic trinomial

Factorising is the reverse of calculating the product of factors. In order to factorise a quadratic, we need to find the factors which, when multiplied together, equal the original quadratic.

Consider a quadratic expression of the form . We see here that is a common factor in both terms. Therefore factorises as . For example, factorises as .

Another type of quadratic is made up of the difference of squares. We know that:

So can be written in factorised form as .

This means that if we ever come across a quadratic that is made up of a difference of squares, we can immediately write down the factors. These types of quadratics are very simple to factorise. However, many quadratics do not fall into these categories and we need a more general method to factorise quadratics.

We can learn about factorising quadratics by looking at the opposite process, where two binomials are multiplied to get a quadratic. For example:

We see that the term in the quadratic is the product of the -terms in each bracket. Similarly, the in the quadratic is the product of the and in the brackets. Finally, the middle term is the sum of two terms.

So, how do we use this information to factorise the quadratic?

Let us start with factorising and see if we can decide upon some general rules. Firstly, write down the two brackets with an in each bracket and space for the remaining terms.

Next, decide upon the factors of . Since the is positive, possible combinations are: 1 and 6, 2 and 3, and or and .

Therefore, we have four possibilities:

Option 1

Option 2

Option 3

Option 4

Next, we expand each set of brackets to see which option gives us the correct middle term.

Option 1

Option 2

Option 3

Option 4

We see that Option 3, , is the correct solution.

The process of factorising a quadratic is mostly trial and error but there are some strategies that can be used to ease the process.

General procedure for factorising a trinomial

Take out any common factor in the coefficients of the terms of the expression to obtain an expression of the form where , and have no common factors and is positive.

Write down two brackets with an in each bracket and space for the remaining terms:

Write down a set of factors for and .

Write down a set of options for the possible factors for the quadratic using the factors of and .

Expand all options to see which one gives you the correct middle term .

If is positive, then the factors of must be either both positive or both negative. If is negative, it means only one of the factors of is negative, the other one being positive. Once you get an answer, always multiply out your brackets again just to make sure it really works.

The following video summarises how to factorise expressions and shows some examples.

Video: 2DKX

Worked example 13: Factorising a quadratic trinomial

Factorise: .

Check that the quadratic is in required form

Write down a set of factors for and

The possible factors for are: 1 and 3

The possible factors for are: and 1

Write down a set of options for the possible factors of the quadratic using the factors of and . Therefore, there are two possible options.

Option 1

Option 2

Check that the solution is correct by multiplying the factors

Write the final answer

Exercise 1.8

Factorise the following:

Sum and difference of two cubes

We now look at two special results obtained from multiplying a binomial and a trinomial:

Sum of two cubes:

Difference of two cubes:

So we have seen that:

We use these two basic identities to factorise more complex examples.

Worked example 14: Factorising a difference of two cubes

Factorise: .

Take the cube root of terms that are perfect cubes

We are working with the difference of two cubes. We know that , so we need to identify and .

We start by noting that and . These give the terms in the first bracket. This also tells us that and .

Find the three terms in the second bracket

We can replace and in the factorised form of the expression for the difference of two cubes with and . Doing so we get the second bracket:

Expand the brackets to check that the expression has been correctly factorised

Worked example 15: Factorising a sum of two cubes

Factorise: .

Take the cube root of terms that are perfect cubes

We are working with the sum of two cubes. We know that , so we need to identify and .

We start by noting that and . These give the terms in the first bracket. This also tells us that and .

Find the three terms in the second bracket

We can replace and in the factorised form of the expression for the sum of two cubes with and . Doing so we get the second bracket:

Expand the brackets to check that the expression has been correctly factorised

Worked example 16: Factorising a difference of two cubes

Factorise: .

Take out the common factor 16

Take the cube root of terms that are perfect cubes

We are working with the difference of two cubes. We know that , so we need to identify and .

We start by noting that and . These give the terms in the first bracket. This also tells us that and .

Find the three terms in the second bracket

We can replace and in the factorised form of the expression for the difference of two cubes with and . Doing so we get the second bracket:

Expand the brackets to check that the expression has been correctly factorised

Worked example 17: Factorising a sum of two cubes

Factorise: .

Take the cube root of terms that are perfect cubes

We are working with the sum of two cubes. We know that , so we need to identify and .

We start by noting that and . These give the terms in the first bracket. This also tells us that and .

Find the three terms in the second bracket

We can replace and in the factorised form of the expression for the difference of two cubes with and . Doing so we get the second bracket:

Expand the brackets to check that the expression has been correctly factorised

Exercise 1.9

Factorise:

Note that .

1.8 Simplification of fractions

We have studied procedures for working with fractions in earlier grades.

Note: dividing by a fraction is the same as multiplying by the reciprocal of the fraction.

In some cases of simplifying an algebraic expression, the expression will be a fraction. For example,

has a quadratic binomial in the numerator and a linear binomial in the denominator. We have to apply the different factorisation methods in order to factorise the numerator and the denominator before we can simplify the expression.

If then the denominator, and the fraction is undefined.

This video shows some examples of simplifying fractions.

Video: 2DNV

Worked example 18: Simplifying fractions

Simplify:

Use grouping to factorise the numerator and take out the common factor in the denominator

Take out common factor in the numerator

Cancel the common factor in the numerator and the denominator to give the final answer

Worked example 19: Simplifying fractions

Simplify:

Factorise the numerator and denominator

Change the division sign and multiply by the reciprocal

Write the final answer

Worked example 20: Simplifying fractions

Simplify:

Factorise the denominators

Make all denominators the same so that we can add or subtract the fractions

The lowest common denominator is .

Write as one fraction

Simplify

Take out the common factor and write the final answer

Worked example 21: Simplifying fractions

Simplify:

Factorise the numerator and denominator

Simplify and find the common denominator

Write the final answer

Exercise 1.10

Simplify (assume all denominators are non-zero):

Note restriction: .

Note restriction: .

Note restriction: .

Note restriction:

Note restriction: .

Note restriction: .

Note restriction: .

Note restrictions: and .

Simplify (assume all denominators are non-zero):

Note restriction: .

Note restriction: .

Note restriction: .

Note restriction: .

Note restrictions: .

Note restrictions: .

Note restrictions: and .

Note restriction: .

Simplify (assume all denominators are non-zero):

Note restrictions: and .

Note restriction: and .

Note restriction: and .

Note restrictions: ; and .

Note restrictions: and .

Note restrictions: and .

Note restriction: .

Note restriction: .

Note restriction: .

Note restriction: .

Note restrictions: ; and .

Note restriction: .

What are the restrictions in the following:

We need to find the value of that will make the denominator equal to . Therefore:

First simplify the fraction:

Now we can determine the restriction:

First simplify the fraction:

Now we can determine the restrictions. There are three restrictions in this case:

Therefore:

1.9 Chapter summary

Presentation: 2DR6

: natural numbers are

: whole numbers are

: integers are

A rational number is any number that can be written as where and are integers and .

The following are rational numbers:

Fractions with both numerator and denominator as integers

Integers

Decimal numbers that terminate

Decimal numbers that recur (repeat)

Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers.

If the root of a number cannot be simplified to a rational number, it is called a surd.

If and are positive whole numbers, and , then .

A binomial is an expression with two terms.

The product of two identical binomials is known as the square of the binomial.

We get the difference of two squares when we multiply

Factorising is the opposite process of expanding the brackets.

The product of a binomial and a trinomial is:

Taking out a common factor is the basic factorisation method.

We often need to use grouping to factorise polynomials.

To factorise a quadratic we find the two binomials that were multiplied together to give the quadratic.

The sum of two cubes can be factorised as:

The difference of two cubes can be factorised as:

We can simplify fractions by incorporating the methods we have learnt to factorise expressions.

Only factors can be cancelled out in fractions, never terms.

To add or subtract fractions, the denominators of all the fractions must be the same.

Exercise 1.11

The figure here shows the Venn diagram for the special sets and .

Where does the number belong in the diagram?

is in its simplest form, therefore it is not in , or . It is in the space between the rectangle and

In the following list, there are two false statements and one true statement. Which of the statements is true?

Every natural number is an integer.

Every whole number is a natural number.

There are fractions in the integers.

Consider each statement:

Integers are natural numbers and negative natural numbers. Therefore this statement is true.

is not a natural number, therefore this statement is false.

Integers are natural numbers and negative natural numbers, no fractions. Therefore this is false.

The only true statement is (i).

State whether the following numbers are real, non-real or undefined.

This is the square root of a negative number and so is non-real.

We are dividing by 0 and so this is undefined.

This is the square root of a positive number and so is real.

This is the square root of a positive number and so is real.

This is the square root of a negative number and so is non-real.

This is the square root of a positive number and so is real.

State whether each of the following numbers are rational or irrational.

Irrational. It cannot be simplified to a fraction of integers.

Irrational. It cannot be simplified to a fraction of integers

Rational. Can be simplified to an integer

Rational. Can be simplified to an integers.

If is an integer, is an integer and is irrational, which of the following are rational numbers?

We have a fraction of integers and so this is rational.

When we divide a number by itself we get 1 and so this is rational.

We are dividing an integer by an irrational number and so this is irrational. However if then the fraction is equal to 0 and the number is rational.

We are dividing an integer by an irrational number and so this is irrational.

Consider the following list of numbers:

Which of the numbers are non-real numbers?

Only is non-real as it is the square root of a negative number.

Without using a calculator, rank all the real numbers in ascending order.

We exclude from the list as it is non-real. We also exclude as it is undefined. Then we note that:

lies between 2 and 8

lies between 6 and 7

lies between and

lies between 8 and 9

Therefore the ordering is:

Which of the numbers are irrational numbers?

Any number that cannot be written as a fraction of integers is irrational. Therefore are all irrational.

Which of the numbers are rational numbers?

All numbers that can be written as a fraction of integers are rational numbers. Therefore are all rational numbers.

Which of the numbers are integers?

Only is an integer.

Which of the numbers are undefined?

Any fraction that has a denominator of 0 is undefined, therefore only is undefined.

Write each decimal as a simple fraction.

Show that the decimal is a rational number.

This is a rational number because both the numerator and denominator are integers.

Write the following fractions as decimal numbers:

Express as a fraction where (show all working).

For each of the following numbers:

write the next three digits;

state whether the number is rational or irrational.

The number does not terminate (this is shown by the ). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

Irrational, there is no repeating pattern.

Since there is a dot over the 1 we know that the 1 repeats. The next three digits are:

Rational, there is a repeating pattern of digits.

Write the following rational numbers to 2 decimal places.

To write to two decimal places we must convert to decimal: .

To write to two decimal places just add a comma and two 's: .

We mark where the cut off point is, determine if it has to be rounded up or not and then write the answer. In this case there is a after the cut off point so we do not round up. The final answer is: .

We mark where the cut off point is, determine if it has to be rounded up or not and then write the answer. In this case there is a after the cut off point so we round up. The final answer is: .

Round off the following irrational numbers to 3 decimal places.

(round up as there is a after the cut off point).

(no rounding as there is a after the cut off point).

(no rounding as there is a after the cut off point).

(no rounding as there is a after the cut off point).

Round off the number to decimal places.

Round off the number to decimal places.

Round off the number to decimal places.

Use your calculator and write the following irrational numbers to 3 decimal places.

Use your calculator (where necessary) and write the following numbers to 5 decimal places. State whether the numbers are irrational or rational.

Irrational number.

Irrational number.

Rational number.

Rational number.

Rational number.

Rational number.

Irrational number.

Rational number.

Rational number.

Irrational number.

Round off:

to the nearest decimal places.

to the nearest decimal places.

Write the following irrational numbers to 3 decimal places and then write each one as a rational number to get an approximation of the irrational number.

Determine between which two consecutive integers the following irrational numbers lie, without using a calculator.

Estimate the following surds to the nearest decimal place, without using a calculator.

lies between 3 and 4. Since and it lies closer to 4 than to 3.

Therefore or are suitable estimates.

lies between 10 and 11. Since and it lies almost exactly between 10 and 11.

Therefore is a suitable estimate.

lies between 6 and 7. Since and it lies closer to 7 than to 6.

Therefore is a suitable estimate.

lies between 7 and 8. Since and it lies almost exactly between the two numbers.

Therefore or are suitable estimates.

Expand the following products:

Expand the following:

What is the value of in ?

From the constant term we see that , therefore .

In :

For which of these values of will be positive?

The term is and so any value greater than will make the term positive.

Therefore

For which of these values of will be positive?

From above we see that the term is . Therefore any positive value of will make positive.

Therefore

For what values of will be positive?

From above we see that the term is . Therefore any positive value of will make positive.

Therefore

For what values of will be positive?

From above we see that any value greater than will make the term positive.

Therefore .

Answer the following:

Expand:

Expand:

Given that , determine the value of without solving for .

Solve by factorising:

Represent the following as a product of its prime factors:

Factorise:

Factorise the following:

Simplify the following:

Show that can be simplified to .

What must be added to to make it equal to ?

Suppose must be added to the expression to get the desired result.

Therefore must be added.

Evaluate if

First simplify the expression:

Now substitute the value of : .

With what expression must be multiplied to get a product of ?

So, the expression is .

With what expression must be divided to get a quotient of ?

Therefore the expression is .

What are the restrictions on the following?

Chapter 2: Exponents

2.1 Introduction

Content covered in this chapter includes the laws of exponents from grade 9 and simplifying expressions with exponents as well as solving simple exponential equations.

The content in this chapter will be used in exponential equations later on as well as in grade 11 for financial calculations.

Note that the rational exponent law is not covered in this chapter, this is only introduced in grade 11.

Exponential notation is a short way of writing the same number multiplied by itself many times. This is very useful in everyday life. You may have heard someone describe the size of an area in square metres or square kilometres. For example, the largest radio telescope in the world is being built in South Africa. The telescope is called the square kilometre array, or SKA. This is because the telescope will occupy an area of kilometre by kilometre or kilometre squared.

Figure 2.1: Antennas from the Square Kilometre Array (artist's concept).

Exponents are also very useful to describe very large and very small numbers. For example, the SKA will be detecting incredibly weak signals from objects which are so far away that to write out the strength of the signal or the number of kilometres away in full would be impractical. Outside of astronomy, exponents are used by many other professions such as computer programmers, engineers, economists, financial analysts, biologists and demographers.

If you want to know more about how exponents are used then take a look at the following presentation.

Presentation: 2DZ9

You have already been introduced to exponents and exponent laws in previous grades. Remember that exponents can also be called indices or powers. Exponential notation is as follows:

For any real number and natural number , we can write multiplied by itself times as: .

Remember the following identities:

Similarly,

Interested in finding out why raising a real number to the power of zero is one? Try work it out for yourself. If you get stuck, you can see an example of how to show this is true at this link.

Video: 2DZB

Look at the following examples to see these identities in action:

If your final answer is easier to work out without a calculator, then write it out in full - not in exponential notation, as in examples 1 and 5.

It is convention to write your final answer with positive exponents.

In this chapter, we will revise the exponent laws and use these laws to simplify and solve more complex expressions and equations.

To revise what exponents are you can watch the following video.

Video: 2DZC

2.2 Revision of exponent laws

There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been done in earlier grades, but we list all the laws here for easy reference:

where , and

The following two videos explain the exponent laws.

Part 1:

Video: 2DZD

Part 2:

Video: 2DZF

Worked example 1: Applying the exponential laws

Simplify:

Worked example 2: Exponential expressions

Simplify:

Change the bases to prime numbers

At first glance it appears that we cannot simplify this expression. However, if we reduce the bases to prime bases, then we can apply the exponent laws.

Simplify the exponents

When you have a fraction that is one term over one term, use the method of Finding Prime Bases - in other words use prime factorisation on the bases.

Worked example 3: Exponential expressions

Simplify:

Change the bases to prime numbers

Subtract the exponents (same base)

Write the answer as a fraction

When working with exponents, all the laws of operation for algebra apply.

Worked example 4: Simplifying by taking out a common factor

Simplify:

Simplify to a form that can be factorised

For each of the exponent laws we can “undo” the law - in other words we can work backwards. For this expression we can reverse the multiplication law to write as .

Take out a common factor

Cancel the common factor and simplify

When you have a fraction that has more than one term in the numerator or denominator, change to prime bases if necessary and then factorise.

Worked example 5: Simplifying using difference of two squares

Simplify:

Change the bases to prime numbers

Factorise using the difference of squares

Cancel the common factor and simplify

Exercise 2.1

Simplify without using a calculator:

2.3 Rational exponents

We can also apply the exponent laws to expressions with rational exponents.

According to CAPS, the rational exponent law is introduced in Grade 11 but you may choose to introduce learners to the rational exponent law at this stage.

Worked example 6: Simplifying rational exponents

Simplify:

Worked example 7: Simplifying rational exponents

Simplify:

Write as a fraction and simplify

Extension: the following video provides a summary of all the exponent rules and rational exponents.

Video: 2F2V

Exercise 2.2

Simplify without using a calculator:

2.5 Exponential equations

Exponential equations have the unknown variable in the exponent. Here are some examples:

If we can write a single term with the same base on each side of the equation, we can equate the exponents. This is one method to solve exponential equations.

Important: if and then:

Also notice that if , then and can be different.

Worked example 8: Equating exponents

Solve for : .

Change the bases to prime numbers

The bases are the same so we can equate exponents

Worked example 9: Equating exponents

Solve for : .

Solve for

We know from the exponent identities that , therefore:

Worked example 10: Solving equations by taking out a common factor

Solve for : .

Rewrite the expression

Take out a common factor

Simplify

Change the bases to prime numbers

The bases are the same so we can equate exponents

Worked example 11: Solving equations by factorising a trinomial

Solve for :

Factorise the trinomial

Solve for

or . However is undefined, so:

Therefore

Worked example 12: Solving equations by factorising a trinomial

Solve for :

Rewrite the equation

We notice that so we can rewrite the equation as:

Factorise as a trinomial

Solve to find both roots

Therefore or .

Learners may find Worked Example 13 much easier using the -substitution method. You may choose to return to this example once the -substitution has been taught.

The solution using -substitution is as follows:

Worked example 13: Solving equations by factorisation

Solve for :

Rewrite the equation

In order to get the equation into a form which we can factorise, we need to rewrite the equation:

Now eliminate the fraction by multiplying both sides of the equation by the denominator, .

Factorise the equation

Now that we have rearranged the equation, we can see that we are left with a difference of two squares. Therefore:

Therefore .

Exercise 2.3

Solve for the variable:

The growth of algae can be modelled by the function . Find the value of such that .

Use trial and error to find the value of correct to 2 decimal places

Use trial and error to find the value of correct to 2 decimal places

Chapter summary

Presentation: 2F44

Exponential notation means writing a number as where is any natural number and is any real number.

is the base and is the exponent or index.

Definition:

, if

, if

, if

The laws of exponents:

When simplifying expressions with exponents, we can reduce the bases to prime bases or factorise.

When solving equations with exponents, we can apply the rule that if then ; or we can factorise the expressions.

Exercise 2.4

Simplify:

Simplify:

Solve:

We check both answers and find that is the only solution.

We check both answers and find that is the only solution.

However no solution exists

Use trial and error to find the value of correct to 2 decimal places

Use trial and error to find the value of correct to 2 decimal places

Explain why the following statements are false:

The sum of two powers of the same degree is not the power of the sum of the bases

The sum of two powers of the same degree is not the power of the sum of the bases

A negative sign is missing, when a power is moved from the denominator to the numerator, the sign of the exponent changes.

From the question we must note that

We cannot multiply bases unless they are raised to the same power

The sign of a base is not changed when an exponent is moved from the denominator to the numerator in a fraction

The power of a product is the product of all the bases raised to the same power

If is written out in full how many digits will there be?

has digits therefore digits.

Prove that

Chapter 3: Number patterns

This chapter covers investigating number patterns that involve a common difference and the general term is linear.

Arithmetic sequences are only covered in grade 12 so do not use here.

The focus of this chapter is more about investigating patterns in numbers and diagrams rather than on formulae.

3.1 Introduction

In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunflower, snowflakes, geometric designs on quilts or tiles, or the number sequence .

Figure 3.1: The pattern of seeds within a sunflower follows the Fibonacci sequence, or

Interested in learning more about the relationship between Fibonacci sequences and sunflowers?

Video: 2F6Y

Try spot any patterns in the following sequences on your own:

Identifying patterns in sequences.

Video: 2F6Z

3.2 Describing sequences

A sequence is an ordered list of items, usually numbers. Each item which makes up a sequence is called a “term”.

Sequences can have interesting patterns. Here we examine some types of patterns and how they are formed.

Examples:

There is difference of between successive terms.

The pattern is continued by adding to the previous term.

There is a difference of between successive terms.

The pattern is continued by adding to (i.e. subtracting from) the previous term.

This sequence has a factor of between successive terms.

The pattern is continued by multiplying the previous term by 2.

This sequence has a factor of between successive terms.

The pattern is continued by multiplying the previous term by .

This sequence has a factor of between successive terms.

The pattern is continued by multiplying the previous term by which is equivalent to dividing the previous term by 3.

Some learners may see example 3 as and see a pattern with the powers. You may choose to discuss this in class as a precursor to geometric series which will be introduced in Grade 12.

Worked example 1: Study table

You and friends decide to study for Maths and are sitting together at a square table. A few minutes later, other friends arrive and would like to sit at your table. You move another table next to yours so that people can sit at the table. Another friends also want to join your group, so you take a third table and add it to the existing tables. Now people can sit together.

Examine how the number of people sitting is related to the number of tables. Is there a pattern?

Two more people can be seated for each table added.

Make a table to see if a pattern forms

Number of tables, n

Number of people seated

n

Describe the pattern

We can see that for tables we can seat people, for tables we can seat people and so on. We started out with people and added two each time. So for each table added, the number of people increased by .

So the pattern formed is .

To describe terms in a number pattern we use the following notation:

The first term of a sequence is .

The fourth term of a sequence is .

The tenth term of a sequence is .

The general term is often expressed as the term and is written as .

A sequence does not have to follow a pattern but, when it does, we can write down the general formula to calculate any term. For example, consider the following linear sequence:

The term is given by the general formula:

You can check this by substituting values into the formula:

If we find the relationship between the position of a term and its value, we can find a general formula which matches the pattern and find any term in the sequence.

Common difference

Consider the following sequence:

We can see that each term is decreasing by 5 but how would we determine the general formula for the term? Let us try to do this with a table.

Term number

Term

Formula

You can see that the difference between the successive terms is always the coefficient of in the formula. This is called a common difference.

Therefore, for sequences with a common difference, the general formula will always be of the form: where is the difference between each term and is some constant.

Sequences with a common difference are called linear sequences.

Common difference

The common difference is the difference between any term and the term before it. The common difference is denoted by .

For example, consider the sequence

To calculate the common difference, we find the difference between any term and the previous term.

Let us find the common difference between the first two terms.

Let us check another two terms:

We see that is constant.

In general,

for example, , not .

Worked example 2: Study table, continued

As before, you and friends are studying for Maths and are sitting together at a square table. A few minutes later other friends arrive so you move another table next to yours. Now people can sit at the table. Another friends also join your group, so you take a third table and add it to the existing tables. Now people can sit together as shown below.

Find an expression for the number of people seated at tables.

Use the general formula to determine how many people can sit around tables.

How many tables are needed to seat people?

Two more people can be seated for each table added.

Make a table to see the pattern

Number of Tables,

Number of people seated

Pattern

n

Note: There may be variations in how you think of the pattern in this problem. For example, you may view this problem as the person on one end fixed, two people seated opposite each other per table and one person at the other end fixed. This results in . Your formula for will still be correct.

Describe the pattern

The number of people seated at tables is

Calculate the term, in other words, find if

Therefore people can be seated at tables.

Calculate the number of tables needed to seat people, in other words, find if

Therefore tables are needed to seat people.

It is important to note the difference between and . can be compared to a place holder indicating the position of the term in the sequence, while is the value of the place held by . From our example above, the first table holds people. So for , the value of and so on:

Worked example 3: Data plans

Raymond subscribes to a limited data plan from Vodacell. The limited data plans cost for gigabyte (GB) per month, for per month and for per month. Assume this pattern continues indefinitely.

Use a table to set up the pattern of the cost of the data plans.

Find the general formula for the sequence.

Use the general formula to determine the cost for a data plan.

The cost of an unlimited data plan is per month. Determine the amount of data Raymond would have to use for it to be cheaper for him to sign up for the unlimited plan.

Make a table to see the pattern

Number of GB

1
2
3
4

Cost (in Rands)

120
135
150
165

Pattern

120

Use the observed pattern to determine the general formula.

The price of GB of data is

Determine the cost of of data.

This question requires us to determine the value of the term, in other words, find if . Using the general formula, we get:

Therefore the cost of a data package is .

Determine when it is cheaper to purchase the unlimited data plan

The final question of this worked example requires us to determine when it would be cheaper for Raymond to purchase an unlimited data plan instead of a limited plan. In other words, we need to find where is less than .

We know that:

Therefore, if

Solving for , we get:

Therefore it is cheaper for Raymond to purchase the unlimited data plan if he uses more than per month.

Learn more about number patterns.

Video: 2F72

Exercise 3.1

Use the given pattern to complete the table below.

Figure number |  |  |  |  |
Number of dots |  |  |  |  |
Number of lines |  |  |  |  |
Total |  |  |  |  |

Figure number |  |  |  |  |
Number of dots |  |  |  |  |
Number of lines |  |  |  |  |
Total |  |  |  |  |

Consider the sequence shown here:

If what is the value of ?

Consider the sequence shown here:

If what is the value of ?

For each of the following sequences determine the common difference. If the sequence is not linear, write “no common difference”.

You can see that the results are not the same - the difference is not 'common.' That means that this sequence of numbers in not linear, and it has no common difference.

All of the results are the same, which means we have found the common difference for these numbers: .

In this case the sequence is not linear. Therefore the final answer is that there is no common difference.

The common difference is .

Write down the next three terms in each of the following sequences:

The common difference is:

Therefore we add each time to get the next term in the sequence. The next three numbers are:

, and

and the sequence becomes:

The common difference is:

Therefore we add each time to get the next term in the sequence. The next three numbers are:

, and

and the sequence becomes:

The common difference is:

Therefore we subtract each time to get the next term in the sequence. The next three numbers are:

, and

and the sequence becomes:

Given a pattern which starts with the numbers: determine the values of and .

Given a sequence which starts with the letters: determine the values of and .

Given a pattern which starts with the numbers: determine the values of and .

The general term is given for each sequence below. Calculate the missing terms (each missing term is represented by ).

The third term is:

The fourth term is:

Therefore the only missing term is the third one, which is . The full sequence is:

The fifth term is:

The sixth term is:

Therefore the only missing term is the fifth one, which is . The full sequence is:

The second term is:

The third term is:

The fourth term is:

The fifth term is:

Therefore the two missing terms are the second and fourth ones, which are and . The full sequence is:

To find the two missing terms, we use the equation for the general term:

Find the general formula for the following sequences and then find , and

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

, and are:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

, and are:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

, and are:

The diagram below shows pictures which follow a pattern.

How many triangles will there be in the picture?

Therefore two triangles are added each time and the fifth picture will have triangles.

Determine the formula for the term.

The general term of the pattern is:

Use the formula to find how many triangles are in the picture of the diagram.

Study the following sequence:

Write down the next terms.

We note that we add 8 to each term to get the next term. Therefore the next three terms are .

Find the general formula for the sequence

Find the value of if is .

Study the following sequence:

Write down the next terms.

We note that we add 30 to each term to get the next term. Therefore the next three terms are .

Find the general formula for the sequence

Find the value of if is .

Consider the following list:

Find the common difference for the terms of the list. If the sequence is not linear (if it does not have a common difference), write “no common difference”.

No common difference.

If you are now told that , determine the values of and .

Consider the following pattern:

Find the common difference for the terms of the pattern. If the sequence is not linear (if it does not have a common difference), write “no common difference”.

The common difference for these numbers: .

If you are now told that , determine the values of and .

If the following terms make a linear sequence: Determine the value of . If the answer is a non-integer, write the answer as a simplified fraction.

Now determine the numeric value of the first three terms. If the answers are not integers, write your answers as fractions.

The first three terms of this sequence are: and .

If the following terms make a linear sequence: find . If the answer is a non-integer, write the answer as a simplified fraction.

Now determine the numeric value of the first three terms. If the answers are not integers, write your answers as fractions.

The first three terms of this sequence are: .

What is the letter of the sequence:

PATTERNPATTERNPATTERNPATTERNPATTERNPATTERNPATTE.............?

The word “PATTERN” is 7 letters long, so:

The remainder of 5 shows us that the letter is the letter in the word, which is E

3.3 Chapter summary

Presentation: 2F85

The general term is expressed as the term and is written as .

We define the common difference of a sequence as the difference between any two successive terms, where

We can work out a general formula for each number pattern and use it to determine any term in the pattern.

Exercise 3.2

Analyse the diagram and complete the table.

Figure number ()

Number of horizontal matches

Number of vertical matches

Total number of matches

Figure number ()

Number of horizontal matches

Number of vertical matches

Total number of matches

Given a list of numbers: determine the common difference for the list (if there is one).

All of the results are the same, which means we have found the common difference for these numbers: .

For the pattern here: calculate the common difference.

If the pattern is not linear, write “no common difference”. Otherwise, give your answer as a decimal.

In this case the sequence is not linear. Therefore the final answer is that there is no common difference.

Consider the list shown here:

If what is the value of ?

Write down the next three terms in each of the following linear sequences:

Given a sequence which starts with the numbers: determine the values of and .

Given a list which starts with the letters: determine the values of and .

Find the sixth term in each of the following sequences:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

is:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

is:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

is:

Find the general formula for the following sequences and then find T₁₀, T₁₅ and T₃₀

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

The general term is given for each sequence below. Calculate the missing terms (each missing term is represented by ).

The missing terms are and

Find the general term in each of the following sequences:

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

We first need to find :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

Study the following sequence

Write down the next terms:

Find the general formula for the sequence.

Find the value of if is .

What is the letter of the sequence:

COMMONCOMMON.............?

The word “COMMON” is 6 letters long, so:

The remainder of 4 shows us that the letter is the letter in the word, which is M

What is the letter of the sequence:

MATHEMATICSMATHEMATICSMATHEMATICSMATHE .............?

The word “MATHEMATICS” is 11 letters long, so:

The remainder of 10 shows us that the letter is the tenth letter in the word, which is C

The seating of a sports stadium is arranged so that the first row has seats, the second row has seats, the third row has seats and so on. Calculate how many seats are in the row.

We start by writing the given information as a sequence:

Now we can calculate :

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

The row is represented by . The number of seats in this row is:

There are 111 seats in the row.

The diagram below shows pictures which follow a pattern.

How many boxes will there be in the sixth picture?

Therefore three boxes are added each time and the sixth picture will have boxes

Determine the formula for the term.

The general term of the pattern is: .

Use the formula to find how many boxes are in the picture of the diagram.

A single square is made from matchsticks. Two squares in a row need matchsticks and three squares need matchsticks.

Answer the following questions for this sequence.

Determine the first term.

We begin by writing a sequence to represent this:

We see from this that the first term is .

Determine the common difference.

The common difference () is:

Determine the general formula.

To determine the general formula we note that for each successive term we add to the last term. We can express this as:

The general formula is .

A row has twenty-five squares. How many matchsticks are there in this row?

We note that a row with twenty-five squares is represented by . The number of matchsticks in this row is:

There are 76 matchsticks in the row with twenty-five squares.

You would like to start saving some money, but because you have never tried to save money before, you decide to start slowly. At the end of the first week you deposit into your bank account. Then at the end of the second week you deposit and at the end of the third week, . After how many weeks will you deposit into your bank account?

We begin by writing down a sequence to represent this:

Next we need to find :

Now we note that for each successive term we add to the last term. We can express this as:

The general formula is .

Now we need to find such that :

After the tenth week you will deposit into your bank account.

Consider the following list:

Find the common difference for the terms of the list. If the sequence is not linear (if it does not have a common difference), write “no common difference”.

The common difference for these numbers: .

If you are now told that , determine the values of and .

If the following terms make a linear sequence: Determine the value of . If the answer is a non-integer, write the answer as a simplified fraction.

Now determine the numeric value of the first three terms. If the answers are not integers, write your answers as fractions.

The first three terms of this sequence are: and .

How many blocks will there be in the picture?

Hint: Use the grey blocks to help.

The grey blocks can be represented by and there are always white blocks.

Analyse the picture below:

How many blocks are there in the next picture?

Picture 1:

Picture 2:

Picture 3:

Picture 4: blocks

Write down the general formula for this pattern.

Look at:

Picture 1:

How many blocks will there be in the 14th picture?

A horizontal line intersects a piece of string at points and divides it into five parts, as shown below.

If the piece of string is intersected in this way by parallel lines, each of which intersects it at points, determine the number of parts into which the string will be divided.

We need to determine a pattern for this scenario.

The first line divides the string into five parts. We can redraw the diagram to show the string with and lines:

From this we see that the two lines cut the string into pieces. Three lines cut the string into pieces. So for each line added we cut the line into more pieces.

So we can write the following sequence:

The common difference is .

Next we note that for each successive term we add to the last term. We can express this as:

The general formula is .

When there are lines we are working with :

Therefore the string will be cut into parts.

Use a calculator to explore and then generalise your findings to determine the:

units digit of

Therefore will follow the same pattern as the third row

therefore the units digit is

tens digit of

Therefore will follow the same pattern as the fourth row

therefore the tens digit is

remainder when is divided by

Therefore will follow the same pattern as the second row

therefore the remainder is

Analyse the diagram and complete the table.

The dots follow a triangular pattern and the formula is .

The general formula for the lines is .

Figure number

Number of dots

Number of lines

Total

We are given the general formula for both the lines and the dots. We can determine the general formula for the sum of the lines and dots by adding the general formula for the lines to the general formula for the dots.

Figure number

Number of dots

Number of lines

Total

Chapter 4: Equations and inequalities

4.1 Introduction

This chapter covers linear, quadratic and simultaneous linear equations as well as word problems, literal equations and linear inequalities.

Linear equations were covered in earlier grades and are revised here.

Word problems can include any of linear, quadratic and simultaneous equations.

For linear inequalities learners must know interval notation and be able to represent the solution graphically.

Equations are widely used to describe the world around us. In science equations are used to describe everything from how a ball rolls down a slope to how the planets move around the sun.

In this chapter we will explore different types of equations as well as looking at how these can be used to solve problems in the real world. We will also look at linear inequalities.

The first use of an “equals” sign from The Whetstone of Witte by Robert Recorde . This equation represents . Recorde is also responsible for introducing the pre-existing “plus” sign (+) to the English-speaking world.

4.2 Solving linear equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the highest exponent of the variable is . The following are examples of linear equations:

Solving an equation means finding the value of the variable that makes the equation true. For example, to solve the simple equation , we need to determine the value of that will make the left hand side equal to the right hand side. The solution is .

The solution, also called the root of an equation, is the value of the variable that satisfies the equation. For linear equations, there is at most one solution for the equation.

To solve equations we use algebraic methods that include expanding expressions, grouping terms, and factorising.

For example:

Check the answer by substituting .

Therefore

The following video gives an introduction to solving linear equations.

Video: 2F9B

Method for solving linear equations

The general steps for solving linear equations are:

Expand all brackets.

Rearrange the terms so that all terms containing the variable are on one side of the equation and all constant terms are on the other side.

Group like terms together and simplify.

Factorise if necessary.

Find the solution and write down the answer.

Check the answer by substituting the solution back into the original equation.

An equation must always be balanced, whatever you do to the left-hand side, you must also do to the right-hand side.

Worked example 1: Solving linear equations

Solve for :

Expand the brackets and simplify

Divide both sides by 10

Check the answer by substituting the solution back into the original equation

Since both sides are equal, the answer is correct.

Worked example 2: Solving linear equations

Solve for :

Multiply both sides of the equation by

Division by is undefined so there must be a restriction: .

Expand the brackets and simplify

Divide both sides by

Check the answer by substituting the solution back into the original equation

Since both sides are equal, the answer is correct.

Worked example 3: Solving linear equations

Solve for :

Multiply the equation by the common denominator and simplify

Rearrange the terms and simplify

Divide both sides by

Check the answer by substituting the solution back into the original equation

Since both sides are equal, the answer is correct.

Exercise 4.1

Solve the following equations (assume all denominators are non-zero):

Note that

Note

4.3 Solving quadratic equations

A quadratic equation is an equation where the exponent of the variable is at most . The following are examples of quadratic equations:

Quadratic equations differ from linear equations in that a linear equation has only one solution, while a quadratic equation has at most two solutions. There are some special situations, however, in which a quadratic equation has either one solution or no solutions.

We solve quadratic equations using factorisation. For example, in order to solve , we need to write it in its equivalent factorised form as . Note that if then or .

The following video shows an example of solving a quadratic equation by factorisation.

Video: 2FBM

Method for solving quadratic equations

Rewrite the equation in the required form, .

Divide the entire equation by any common factor of the coefficients to obtain an equation of the form , where , and have no common factors. For example can be written as .

Factorise to be of the form .

The two solutions are or , so or , respectively.

Check the answer by substituting it back into the original equation.

Worked example 4: Solving quadratic equations

Solve for :

The equation is already in the required form,

Factorise

Solve for both factors

We have

OR

Check both answers by substituting back into the original equation

Write the final answer

The solution to is or .

Worked example 5: Solving quadratic equations

Find the roots:

The equation is already in the required form,

Divide the equation by common factor

Factorise

The quadratic is a perfect square

This is an example of a special situation in which there is only one solution to the quadratic equation because both factors are the same.

Check the answer by substituting back into the original equation

Write final answer

The solution to is .

Exercise 4.2

Write the following in standard form

Solve the following equations:

Solve the following equations (note the restrictions that apply):

Note

Note

Note

Note

Note

Note

Note that has no real solutions.

Note

Note

Note

4.4 Solving simultaneous equations

Up to now we have solved equations with only one unknown variable. When solving for two unknown variables, two equations are required and these equations are known as simultaneous equations. The solutions are the values of the unknown variables which satisfy both equations simultaneously. In general, if there are unknown variables, then independent equations are required to obtain a value for each of the variables.

An example of a system of simultaneous equations is:

We have two independent equations to solve for two unknown variables. We can solve simultaneous equations algebraically using substitution and elimination methods. We will also show that a system of simultaneous equations can be solved graphically.

Solving by substitution

Use the simplest of the two given equations to express one of the variables in terms of the other.

Substitute into the second equation. By doing this we reduce the number of equations and the number of variables by one.

We now have one equation with one unknown variable which can be solved.

Use the solution to substitute back into the first equation to find the value of the other unknown variable.

The following video shows how to solve simultaneous equations using substitution.

Video: 2FD5

Worked example 6: Simultaneous equations

Solve for and :

Use equation to express in terms of

Substitute into equation and solve for

Substitute back into equation and solve for

Check the solution by substituting the answers back into both original equations

Write the final answer

Worked example 7: Simultaneous equations

Solve the following system of equations:

Use either equation to express in terms of

Substitute into equation and solve for

Substitute back into equation and solve for

Check the solution by substituting the answers back into both original equations

Write the final answer

Solving by elimination

Worked example 8: Simultaneous equations

Solve the following system of equations:

Make the coefficients of one of the variables the same in both equations

The coefficients of in the given equations are and . Eliminate the variable by adding equation and equation together:

Simplify and solve for

Substitute back into either original equation and solve for

Check that the solution and satisfies both original equations

Write the final answer

Worked example 9: Simultaneous equations

Solve the following system of equations:

Make the coefficients of one of the variables the same in both equations

By multiplying equation by and equation by , both coefficients of will be .

(When subtracting two equations, be careful of the signs.)

Simplify and solve for

Substitute value of back into either original equation and solve for

Check that the solution and satisfies both original equations

Write the final answer

Solving graphically

This section can be included in the chapter on functions and graphs with graphs of linear equations. Before beginning this section it may be necessary to revise plotting graphs of linear equations with your learners.

It is also important that learners are either given the graphs or are encouraged to draw accurate graphs on graph paper to help them solve simultaneous equations graphically. Graph sketching software can be used in this section to ensure that graphs are accurate.

Simultaneous equations can also be solved graphically. If the graphs of each linear equation are drawn, then the solution to the system of simultaneous equations is the coordinates of the point at which the two graphs intersect.

For example:

The graphs of the two equations are shown below.

The intersection of the two graphs is . So the solution to the system of simultaneous equations is and . We can also check the solution using algebraic methods.

Substitute equation into :

Then solve for :

Substitute the value of back into equation :

Notice that both methods give the same solution.

You can use an online tool such as graphsketch to draw the graphs and check your solution.

Worked example 10: Simultaneous equations

Solve the following system of simultaneous equations graphically:

Write both equations in form

Sketch the graphs on the same set of axes

Find the coordinates of the point of intersection

The two graphs intersect at

Write the final answer

Exercise 4.3

Look at the graph below

Solve the equations and simultaneously

From the graph we can see that the lines intersect at and

Look at the graph below

Solve the equations and simultaneously

The lines are parallel. Therefore there is no solution to and .

Look at the graph below

Solve the equations and simultaneously

From the graph we can see that the lines intersect at and

Solve for and :

and .

Solve for :

Substitute the value of into the second equation and solve for :

Therefore .

and

Write in terms of :

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

Write in terms of :

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

Write in terms of :

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

Make the subject of the first equation:

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

Make the subject of the first equation:

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

If we multiply the first equation by 2 then the coefficient of will be the same in both equations:

Now we can subtract the second equation from the first:

Solve for :

Substitute the value of into the first equation and solve for :

Therefore .

and

Make the subject of the first equation:

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore .

and

If we subtract the second equation from the first then we can solve for :

Solve for :

Substitute the value of into the first equation and solve for :

Therefore .

and

Make the subject of the first equation:

Substitute the value of into the second equation and solve for :

Substitute the value of in the first equation:

Therefore .

and

Rearrange both equations by multiplying by :

Add the two equations together:

Solve for :

Substitute value of back into first equation:

Therefore .

and

Let

Since this is true for all in the real numbers, can be any real number.

Look at what happens to when is very small or very large:

The smallest can be is 0. When , .

If gets very large, then the fraction becomes very small (think about what happens when you divide a small number by a very large number). Then .

From this we can see that .

Therefore can be any real number, .

and

Note

Look at the first equation

Note that this is the same as the second equation

and can be any real number except for .

Solve graphically and check your answer algebraically:

and

First write the equations in standard form:

Draw the graph:

The graphs intersect at so and .

Checking algebraically we get:

Substitute value of into second equation:

Substitute the value of back into the first equation:

and

First write the equations in standard form:

Draw the graph:

The graphs intersect at so and .

Checking algebraically we get:

Substitute value of into first equation:

Substitute the value of back into the first equation:

and

First write the equations in standard form:

Draw the graph:

The graphs intersect at so and .

Checking algebraically we get:

Substitute value of into first equation:

Substitute the value of back into the first equation:

and

First write the equations in standard form:

Draw the graph:

The graphs intersect at so and .

Checking algebraically we get:

Substitute value of into first equation:

Substitute the value of back into the first equation:

and

First write the equations in standard form:

Draw the graph:

The graphs intersect at so and .

Checking algebraically we get:

Substitute value of into second equation:

Substitute the value of back into the first equation:

4.5 Word problems

To solve word problems we need to write a set of equations that represent the problem mathematically. The solution of the equations is then the solution to the problem.

Problem solving strategy

Read the whole question.

What are we asked to solve for?

Assign a variable to the unknown quantity, for example, .

Translate the words into algebraic expressions by rewriting the given information in terms of the variable.

Set up an equation or system of equations to solve for the variable.

Solve the equation algebraically using substitution.

Check the solution.

The following video shows two examples of working with word problems.

Video: 2FDX

Worked example 11: Solving word problems

A shop sells bicycles and tricycles. In total there are cycles (cycles include both bicycles and tricycles) and wheels. Determine how many of each there are, if a bicycle has two wheels and a tricycle has three wheels.

Assign variables to the unknown quantities

Let be the number of bicycles and let be the number of tricycles.

Set up the equations

Rearrange equation and substitute into equation

Calculate the number of tricycles

Write the final answer

There are tricycles and bicycles.

Worked example 12: Solving word problems

Bongani and Jane are friends. Bongani takes Jane's maths test paper and will not tell her what her mark is. He knows that Jane dislikes word problems so he decides to tease her. Bongani says: “I have marks more than you do and the sum of both our marks is equal to . What are our marks?”

Assign variables to the unknown quantities

We have two unknown quantities, Bongani's mark and Jane's mark. Let Bongani's mark be and Jane's mark be .

Set up a system of equations

Bongani has more marks than Jane.

Both marks add up to .

Use equation to express in terms of

Substitute into equation

Rearrange and solve for

Substitute the value for back into equation and solve for

Check that the solution satisfies both original equations

Write the final answer

Bongani got for his test and Jane got .

Worked example 13: Solving word problems

A fruitshake costs more than a chocolate milkshake. If fruitshakes and chocolate milkshakes cost , determine the individual prices.

Assign variables to the unknown quantities

Let the price of a chocolate milkshake be and let the price of a fruitshake be .

Set up a system of equations

Substitute equation into

Rearrange and solve for

Substitute the value of back into equation and solve for

Check that the solution satisfies both original equations

Write final answer

One chocolate milkshake costs and one fruitshake costs .

Worked example 14: Solving word problems

The product of two consecutive negative integers is . Find the two integers.

Assign variables to the unknown quantities

Let the first integer be and let the second integer be

Set up an equation

Expand and solve for

Find the sign of the integers

It is given that both integers must be negative.

Write the final answer

The two consecutive negative integers are and .

Exercise 4.4

Two jets are flying towards each other from airports that are apart. One jet is flying at and the other jet at . If they took off at the same time, how long will it take for the jets to pass each other?

Let distance and distance .

Speed and speed .

Time is found by dividing distance by speed.

When the jets pass each other:

Now we know the distance travelled by the second jet when it passes the first jet, we can find the time:

It will take take the jets 2 hours to pass each other.

Two boats are moving towards each other from harbours that are apart. One boat is moving at and the other boat at . If both boats started their journey at the same time, how long will they take to pass each other?

Notice that the sum of the distances for the two boats must be equal to the total distance when the boats meet: .

This question is about distances, speeds, and times. The equation connecting these values is

You want to know the amount of time needed for the boats to meet - let the time taken be . Then you can write an expression for the distance each of the boats travels:

Now we can substitute the two expressions for the distances into the expression for the total distance:

The boats will meet after hour.

Zwelibanzi and Jessica are friends. Zwelibanzi takes Jessica's civil technology test paper and will not tell her what her mark is. He knows that Jessica dislikes word problems so he decides to tease her. Zwelibanzi says: “I have marks more than you do and the sum of both our marks is equal to . What are our marks?”

Let Zwelibanzi's mark be and let Jessica's mark be . Then

Substitute the first equation into the second equation and solve:

Substituting this value back into the first equation gives: Zwelibanzi achieved marks and Jessica achieved marks.

Kadesh bought shirts at a total cost of . If the large shirts cost and the small shirts cost , how many of each size did he buy?

Let be the number of large shirts and the number of small shirts.

Next we note the following:

He bought large shirts for

He bought small shirts for

He spent in total

We can represent the cost as:

Therefore Kadesh buys large shirts and small shirts.

The diagonal of a rectangle is more than its width. The length of the rectangle is more than its width. What are the dimensions of the rectangle?

Let length , width and diagonal . and .

By the theorem of Pythagoras:

The width must be positive, therefore: width length and diagonal .

The sum of and is equal to more than an unknown number. Find the unknown number.

Let the unknown number .

The unknown number is .

A group of friends is buying lunch. Here are some facts about their lunch:

a milkshake costs more than a wrap

the group buys 8 milkshakes and 2 wraps

the total cost for the lunch is

Determine the individual prices for the lunch items.

Let a milkshake be and a wrap be . From the given information we get the following equations:

Substitute the first equation into the second equation and solve for :

Substitute the value of into the first equation and solve for :

Therefore a milkshake costs and a wrap costs .

The two smaller angles in a right-angled triangle are in the ratio of . What are the sizes of the two angles?

Let the smallest angle. Therefore the other angle .

We are given the third angle .

The sizes of the angles are and .

The length of a rectangle is twice the breadth. If the area is , determine the length and the breadth.

We are given length and .

Substitute the first equation into the second equation and solve for :

But breadth must be positive, therefore .

Substitute this value into the first equation to solve for :

Therefore and .

If times a number is increased by , the result is less than the square of the number. Find the number.

Let the number . The equation that expresses the given information is:

We are not told if the number is positive or negative. Therefore the number is or .

The length of a rectangle is more than the width of the rectangle. The perimeter of the rectangle is . Find the length and the width of the rectangle.

Let length , width and perimeter .

and .

length: , width:

Stephen has 1 litre of a mixture containing % salt. How much water must Stephen add to make the mixture % salt? Write your answer as a fraction of a litre.

The new volume () of mixture must contain % salt, therefore:

The volume of the new mixture is litre The amount of water () to be added is:

Therefore litres of water must be added. To write this as a fraction of a litre:

Therefore must be added.

The sum of two consecutive odd numbers is and their difference is . Find the two numbers.

Let the numbers be and .

Then the two equations describing the constraints are:

Add the first equation to the second equation:

Substitute into first equation:

Therefore the two numbers are 9 and 11.

The denominator of a fraction is more than the numerator. The sum of the fraction and its reciprocal is . Find the fraction.

Let the numerator be . So the denominator is .

Solve for :

From this the fraction could be or . For the second solution we can simplify the fraction to and in this case the denominator is not 1 less than the numerator.

So the fraction is .

Masindi is years older than her daughter, Mulivhu. The sum of their ages is . How old is Mulivhu?

Let Mulivhu be years old. So Masindi is years old.

Mulivhu is years old.

Tshamano is now five times as old as his son Murunwa. Seven years from now, Tshamano will be three times as old as his son. Find their ages now.

Let Murunwa be years old. So Tshamano is years old.

In years time Murunwa's age will be . Tshamano's age will be .

So Murunwa is 7 years old and Tshamano is 35 years old.

and years old.

If adding one to three times a number is the same as the number, what is the number equal to?

Let the number be . Then:

If a third of the sum of a number and one is equivalent to a fraction whose denominator is the number and numerator is two, what is the number?

Let the number be . Then:

Rearrange until we get a trinomial and solve for :

A shop owner buys 40 sacks of rice and mealie meal worth in total. If the rice costs per sack and mealie meal costs per sack, how many sacks of mealie meal did he buy?

There are 100 bars of blue and green soap in a box. The blue bars weigh per bar and the green bars per bar. The total mass of the soap in the box is . How many bars of green soap are in the box?

Lisa has 170 beads. She has blue, red and purple beads each weighing , and respectively. If there are twice as many red beads as there are blue beads and all the beads weigh , how many beads of each type does Lisa have?

4.6 Literal equations

A literal equation is one that has several letters or variables. Examples include the area of a circle and the formula for speed . In this section we solve literal equations in terms of one variable. To do this, we use the principles we have learnt about solving equations and apply them to rearranging literal equations. Solving literal equations is also known as changing the subject of the formula.

Keep the following in mind when solving literal equations:

We isolate the unknown variable by asking “what is it joined to?” and “how is it joined?” We then perform the opposite operation to both sides as a whole.

If the unknown variable is in two or more terms, then we take it out as a common factor.

If we have to take the square root of both sides, remember that there will be a positive and a negative answer.

If the unknown variable is in the denominator, we multiply both sides by the lowest common denominator (LCD) and then continue to solve.

The following video shows an example of solving literal equations.

Video: 2FFQ

Worked example 15: Solving literal equations

The area of a triangle is . What is the height of the triangle in terms of the base and area?

Isolate the required variable

We are asked to isolate the height, so we must rearrange the equation with on one side of the equals sign and the rest of the variables on the other.

Write the final answer

The height of a triangle is given by:

Worked example 16: Solving literal equations

Given the formula: make the subject of the formula.

Isolate the required variable

Exercise 4.5

Solve for in the following formula: .

Make the subject of the formula: .

Note restriction:

Solve for : .

Note restrictions: ,

Make the subject of the formula: .

Note restriction:

Solve for : .

Note restriction:

Solve for : .

Note restriction:

Solve for : .

Note restriction:

Make the subject of the formula: .

Note restrictions: ,

Solve for : .

Note restriction:

Solve for : .

Solve for : .

Make the subject of the formula: .

Note restriction:

Solve for : .

is the formula for converting temperature in degrees Celsius to degrees Fahrenheit. Derive a formula for converting degrees Fahrenheit to degrees Celcius.

To convert degrees Fahrenheit to degrees Celsius we use:

is the formula for determining the volume of a soccer ball. Express the radius in terms of the volume.

Therefore expressing the radius in terms of the volume gives:

Solve for in:

Solve for in:

Solve for in if , ,

Solve for in if , ,

Solve for in if ,

Solve for in if ,

4.7 Solving linear inequalities

A linear inequality is similar to a linear equation in that the largest exponent of a variable is . The following are examples of linear inequalities.

The methods used to solve linear inequalities are similar to those used to solve linear equations. The only difference occurs when there is a multiplication or a division that involves a minus sign. For example, we know that . If both sides of the inequality are divided by , then we get , which is not true. Therefore, the inequality sign must be switched around, giving .

In order to compare an inequality to a normal equation, we shall solve an equation first.

Solve :

If we represent this answer on a number line, we get:

Now let us solve for in the inequality :

If we represent this answer on a number line, we get:

We see that for the equation there is only a single value of for which the equation is true. However, for the inequality, there is a range of values for which the inequality is true. This is the main difference between an equation and an inequality.

Remember: when we divide or multiply both sides of an inequality by a negative number, the direction of the inequality changes. For example, if , then . Also note that we cannot divide or multiply by a variable.

The following video provides an introduction to linear inequalities.

Video: 2FGH

Interval notation

Examples:

Round brackets indicate that the number is not included. This interval includes all real numbers greater than but not equal to 4 and less than but not equal to 12.

Round brackets are always used for positive and negative infinity. This interval includes all real numbers less than, but not equal to .

A square bracket indicates that the number is included. This interval includes all real numbers greater than or equal to 1 and less than but not equal to 13.

It is important to note that this notation can only be used to represent an interval of real numbers.

We represent the above answer in interval notation as

Worked example 17: Solving linear inequalities

Solve for :

Represent the answer on a number line and in interval notation.

Rearrange and solve for

Multiply by and reverse inequality sign

Represent the answer on a number line

Represent the answer in interval notation

Worked example 18: Solving linear inequalities

Solve for :

Represent the answer on a number line and in interval notation.

Expand the bracket

Rearrange and solve for

Divide both sides by

Represent the answer on a number line

Represent the answer in interval notation

Worked example 19: Solving compound linear inequalities

Solve for :

Represent the answer on a number line and in interval notation.

Subtract from all the parts of the inequality

Represent the answer on a number line

Represent the answer in interval notation

Exercise 4.6

Look at the number line and write down the inequality it represents.

Solve for and represent the answer on a number line and in interval notation.

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

The inequality is true for all real values of .

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Represented on a number line:

In interval notation:

Solve for and show your answer in interval notation:

.

.

The interval is:

The interval is:

Solve the inequality:

Solve the inequality:

Solve for the unknown variable and show your answer on a number line.

However we are told that and so .

4.8 Chapter summary

Presentation: 2FHK

A linear equation is an equation where the exponent of the variable is . A linear equation has at most one solution.

A quadratic equation is an equation where the exponent of the variable is at most . A quadratic equation has at most two solutions.

To solve for two unknown variables, two equations are required. These equations are known as a system of simultaneous equations. There are two ways to solve linear simultaneous equations: algebraic solutions and graphical solutions. To solve algebraically we use substitution or elimination methods. To solve graphically we draw the graph of each equation and the solution will be the coordinates of the point of intersection.

Literal equations are equations that have several letters and variables.

Word problems require a set of equations that represent the problem mathematically.

A linear inequality is similar to a linear equation and has the exponent of the variable equal to .

If we divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes.

Exercise 4.7

Solve:

Note that

Solve:

Note restrictions: .

Note restrictions: .

Look at the graph below:

Solve the equations and simultaneously.

From the graph we can see that the lines intersect at and

Look at the graph below:

Solve the equations and simultaneously.

The lines are parallel therefore there is no solution to and .

Look at the graph below:

Solve the equations and simultaneously.

From the graph we can see that the lines intersect at and

Solve the following simultaneous equations:

and

Add the two equations to remove the term and solve for :

Therefore .

Substitute value of into second equation:

Therefore .

and

Substitute value of into first equation:

Substitute value of back into second equation:

Therefore .

and

Make the subject of the first equation:

Substitute value of into first equation:

Substitute value of back into second equation:

Therefore .

and

Make the subject of the first equation:

Substitute value of into second equation:

Substitute value of back into first equation:

Therefore and .

and

Multiply the first equation by 2 so that the coefficient of is the same as the second equation:

Subtract the second equation from the first equation:

Solve for :

Substitute the value of into the first equation and solve for :

Therefore and .

and

Multiply the first equation by 4 so that the coefficient of is the same as the second equation:

Subtract the second equation from the first equation:

Solve for :

Substitute the value of into the first equation and solve for :

Therefore and .

and

Write the first equation in terms of :

Substitute the value of into the second equation:

Substitute the value of into the first equation and solve for :

Therefore and .

and

Make the subject of the second equation:

Substitute the value of into the first equation:

Substitute the value of into the second equation:

Therefore and .

and

Multiply the first equation by 4 so that the coefficient of is the same as the second equation:

Subtract the second equation from the first equation:

Solve for :

Substitute the value of into the first equation and solve for :

Therefore and .

and

Multiply the first equation by 5 so that the coefficient of is the same as the second equation:

Subtract the second equation from the first equation:

Solve for :

Substitute the value of into the first equation and solve for :

Therefore and .

and

Make the subject of the second equation:

Substitute the value of into the first equation:

Substitute the value of into the second equation and solve for :

Therefore and .

and

Make the subject of the second equation:

Substitute the value of into the first equation:

Substitute the value of into the second equation and solve for :

Therefore and .

and

We note that the second equation has a common factor of 2:

Now we can subtract the second equation from the first:

There is no solution for this system of equations. We can see this if we graph the two equations:

From the graph we see that the lines have the same gradient and do not intersect.

Therefore there is no solution.

and

Look at the first equation

Note that this is the same as the second equation

and can be any real number except for .

and

Look at the second equation:

Note that this is the same as the first equation.

and can be any real number except for .

and

Note that and

Make the subject of equation 1:

Make the subject of equation 2:

Substitute equation 3 into equation 4:

Substitute the value of into equation 3:

Therefore

Note that and are both greater than or equal to zero therefore for the equation to be true they must both equal zero.

Find the solutions to the following word problems:

of a certain number is more than of the number. Find the number.

Let be the number.

The number is .

Three rulers and two pens have a total cost of . One ruler and one pen have a total cost of . How much does a ruler cost and how much does a pen cost?

Let the price of a ruler be and the price of a pen be .

From the second equation:

Substitute the value of into the first equation:

Substitute the value of into the second equation:

Therefore each ruler costs and each pen costs .

A group of friends is buying lunch. Here are some facts about their lunch:

a hotdog costs more than a milkshake

the group buys 3 hotdogs and 2 milkshakes

the total cost for the lunch is

Determine the individual prices for the lunch items.

Let the price of a hotdog be and the price of a milkshake be . From the given information we get:

Substitute the first equation into the second equation:

Substitute the value of into the first equation:

The price of the hotdog is while a milkshake costs .

Lefu and Monique are friends. Monique takes Lefu's business studies test paper and will not tell him what his mark is. She knows that Lefu dislikes word problems so she decides to tease him. Monique says: “I have marks more than you do and the sum of both our marks is equal to . What are our marks?”

Let Lefu's mark be and let Monique's mark be . Then

Substitute the first equation into the second equation and solve:

Substituting this value back into the first equation gives:

The students obtained the following marks: Lefu has marks and Monique has marks.

A man runs to the bus stop and back in minutes. His speed on the way to the bus stop is and his speed on the way back is . Find the distance to the bus stop.

Let be the distance to the bus stop.

Speed and .

Distance is given by speed times time. The man runs the same distance to the bus stop as he does from the bus stop. Therefore:

He takes a total of 15 minutes to run there and back so the total time is . However the speeds are given in kilometers per hour and so we must convert the time to hours. Therefore

Next we note that and .

Therefore:

The bus stop is away.

Two trucks are travelling towards each other from factories that are apart. One truck is travelling at and the other truck at . If both trucks started their journey at the same time, how long will they take to pass each other?

Notice that the sum of the distances for the two trucks must be equal to the total distance when the trucks meet: .

This question is about distances, speeds and times. The equation connecting these values is

You want to know the amount of time needed for the trucks to meet - let the time taken be . Then you can write an expression for the distance each of the trucks travels:

Now you have three different equations: you must solve them simultaneously; substitution is the easiest choice. The trucks will meet after hours.

Zanele and Piet skate towards each other on a straight path. They set off apart. Zanele skates at and Piet at . How far will Piet have skated when they reach each other?

Let be the distance that Zanele skates and the distance Piet skates.

Next we note the following information:

Zanele skates at a speed of in a time of

Piet skates at a speed of in a time of

Zanele will have skated and Piet will have skated when they reach other.

When the price of chocolates is increased by , we can buy five fewer chocolates for . What was the price of each chocolate before the price was increased?

Let be the original price of chocolates. The new price of chocolates is .

Since price has to be positive the chocolates used to cost .

A teacher bought worth of textbooks. The text books were for Science and Mathematics with each of them being sold at per book and per book respectively. If the teacher bought 97 books in total, how many Science books did she buy?

She bought science books.

Thom's mom bought worth of easter eggs. The easter eggs came in 3 different colours blue, green and yellow. The blue ones cost each, green ones each and yellow ones each. She bought three times as many yellow eggs as the green ones and eggs in total. How many blue eggs did she buy?

Two equivalent fractions both have their numerator as one. The denominator of one fraction is the sum of two and a number, while the other fraction is twice the number less 3. What is the number?

Consider the following literal equations:

Solve for :

Solve for : .

Note restriction: .

Make the subject of the formula: .

Note restriction: .

Solve for : .

Note restriction: .

Make the subject of the formula: .

Note restriction: .

Solve for : .

Solve for in:

Solve for in if and

Solve for in if , ,

Write down the inequality represented by the following:

Solve for and show your answer in interval notation

Now solve. (Remember to flip the inequality symbol if you multiply or divide by a negative.)

.

The interval is:

Solve the inequality:

Solve and represent your answer on a number line

Solve for the unknown variable

Note

Note

Note

Note

Note

Note that .

Note that you cannot simply divide through by , because it is not stated that , so the value of may be and you cannot divide by zero.

Note that in this solution and are denominators, this means that they are no equal to zero.

Note that and , because the numerator cannot be zero

After solving an equation, Luke gave his answer as rounded to one decimal digit. Show on a number line the interval in which his solution lay.

Chapter 5: Trigonometry

5.1 Introduction

Content covered in this chapter includes defining the trigonometric ratios and extending these definitions to any angle. Also covered is the definitions of the reciprocals of the trigonometric ratios. Both the trigonometric ratios and their reciprocals are solved for several special angles. In addition simple trigonometric equations are covered.

Solving problems in two-dimensions using trigonometry is only covered later in the year and the content for this can be found in chapter 11.

Similarity of triangles is fundamental to the trigonometric ratios

Trigonometric ratios are independent of the lengths of the sides and instead depend only on the angles

Doubling a ratio has a different effect from doubling an angle.

Emphasise the value and importance of making sketches, where appropriate.

Remind learners that angles in the Cartesian plane are always measured from the positive -axis.

When working with angles on the Cartesian plane remind learners to check that their answers are within the correct quadrant.

Calculator skills are very important in this chapter. Methods for CASIO calculators are shown but practical demonstration may be required. For a SHARP calculator the keys are generally the same although the \) key is now the \) key.

We will refer to sine, cosine, tangent, secant, cosecant and cotangent as trigonometric ratios rather than as trigonometric functions. Both these terms are correct though but for the nature of the content in this chapter the term ratio better captures the content and is likely to be more accessible to learners at this stage.

Trigonometry deals with the relationship between the angles and sides of a triangle. We will learn about trigonometric ratios in right-angled triangles, which form the basis of trigonometry.

There are many applications of trigonometry. Of particular value is the technique of triangulation, which is used in astronomy to measure the distances to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. GPS (the global positioning system) would not be possible without trigonometry. Other fields which make use of trigonometry include acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), chemistry, cryptology, meteorology, oceanography, land surveying, architecture, phonetics, engineering, computer graphics and game development.

Figure 5.1: An artist's depiction of a GPS satellite orbiting the Earth. There are at least 24 GPS satellites operational at any one time. GPS uses an application of trigonometry, known as triangulation, to determine ones position. The accuracy of GPS is to within 15 metres.

Fabumaths has some useful links and content for trigonometry.

The following video covers a brief history of trigonometry and some of the uses of trigonometry.

Video: 2FN7

5.2 Similarity of triangles

Before we delve into the theory of trigonometry, complete the following investigation to get a better understanding of the foundation of trigonometry.

Ratios of similar triangles

Draw three similar triangles of different sizes using a protractor and a ruler, with each triangle having interior angles equal to °, ° and ° as shown below. Measure the angles and lengths accurately in order to fill in the table (leave your answers as a simplified fraction):

Dividing lengths of sides (ratios)

What observations can you make about the ratios of the sides?

Have you noticed that it does not matter what the lengths of the sides of the triangles are, if the angle remains constant, the ratio of the sides will always yield the same answer?

In the triangles below, is similar to . This is written as:

In similar triangles, it is possible to deduce ratios between corresponding sides:

Another important fact about similar triangles and is that the angle at vertex is equal to the angle at vertex , the angle at vertex is equal to the angle at vertex , and the angle at vertex is equal to the angle at vertex .

The order of letters for similar triangles is very important. Always label similar triangles in corresponding order. For example,

5.3 Defining the trigonometric ratios

The ratios of similar triangles are used to define the trigonometric ratios. Consider a right-angled triangle with an angle marked (said 'theta').

In a right-angled triangle, we refer to the three sides according to how they are placed in relation to the angle . The side opposite to the right-angle is labelled the hypotenuse, the side opposite is labelled “opposite”, the side next to is labelled “adjacent”.

You can choose either non-° internal angle and then define the adjacent and opposite sides accordingly. However, the hypotenuse remains the same regardless of which internal angle you are referring to because it is always opposite the right-angle and always the longest side.

We define the trigonometric ratios: sine (), cosine () and tangent (), of an angle, as follows:

These ratios, also known as trigonometric identities, relate the lengths of the sides of a right-angled triangle to its interior angles. These three ratios form the basis of trigonometry.

The definitions of opposite, adjacent and hypotenuse are only applicable when working with right-angled triangles! Always check to make sure your triangle has a right-angle before you use them, otherwise you will get the wrong answer.

You may also hear people saying “Soh Cah Toa”. This is a mnemonic technique for remembering the trigonometric ratios:

The following video introduces the three trigonometric ratios.

Video: 2FN8

Worked example 1: Trigonometric ratios

Given the following triangle:

Label the hypotenuse, opposite and adjacent sides of the triangle with respect to .

State which sides of the triangle you would use to find , and .

Label the hypotenuse, opposite and adjacent sides of the triangle with respect to .

State which sides of the triangle you would use to find , and .

Label the triangle

First find the right angle, the hypotenuse is always directly opposite the right angle. The hypotenuse never changes position, it is always directly opposite the right angle and so we find this first.

The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite (as the word opposite suggests) the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

Complete the trigonometric ratios

Now we can complete the trigonometric ratios for :

Therefore to find we would use sides (opposite side to ) and (hypotenuse). To find we would use sides (adjacent side to ) and (hypotenuse). To find we would use sides (opposite side to ) and (adjacent side to ).

And then we can complete the trigonometric ratios for . For angle the opposite and adjacent sides switch places (redraw the triangle above to help you see this). Notice how the hypotenuse is still .

Therefore to find we would use sides (opposite side to ) and (hypotenuse). To find we would use sides (adjacent side to ) and (hypotenuse). To find we would use sides (opposite side to ) and (adjacent side to ).

Exercise 5.1

Complete each of the following:

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

In each of the following triangles, state whether , and are the hypotenuse, opposite or adjacent sides of the triangle with respect to .

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the adjacent side

is the hypotenuse

is the opposite side

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the adjacent side

is the opposite side

is the hypotenuse

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the opposite side

is the adjacent side

is the hypotenuse

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the hypotenuse

is the opposite side

is the adjacent side

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the opposite side

is the hypotenuse

is the adjacent side

First find the right angle, the hypotenuse is always directly opposite the right angle. The opposite and adjacent sides depend on the angle we are looking at. The opposite side relative to the angle is directly opposite the angle . Finally the adjacent side is the remaining side of the triangle and must be one of the sides that forms the angle .

is the adjacent side

is the hypotenuse

is the opposite side

Consider the following diagram:

Without using a calculator, answer each of the following questions.

Write down in terms of , and .

is the adjacent side

is the hypotenuse

is the opposite side

Write down in terms of , and .

is the opposite side

is the hypotenuse

is the adjacent side

Write down in terms of , and .

is the adjacent side

is the hypotenuse

is the opposite side

Write down in terms of , and .

is the opposite side

is the hypotenuse

is the adjacent side

Find in the diagram in three different ways. You do not need to calculate the value of , just write down the appropriate ratio for .

Side of length 4 is the opposite side

Side of length 5 is the hypotenuse

Side of length 3 is the adjacent side

Notice that the hypotenuse is the longest side as we would expect.

Which of these statements is true about ?

We first find the opposite and adjacent sides with respect to and :

is the opposite side to and the adjacent side to

is the hypotenuse

is the adjacent side to and the opposite side to

We also note that:

Looking at each of the given ratios we can see that only is correct.

Sarah wants to find the value of in the triangle below. Which statement is a correct line of working?

Sarah first needs to identify the hypotenuse, opposite and adjacent sides in the triangle. She then needs to write down a trigonometric ratio that will allow her to find .

is one such ratio that will help her find . From the given list of options this is the only correct line of reasoning.

has the angle and the lengths of the sides switched around.

uses the wrong sides with respect to for .

uses the wrong sides with respect to for . Note that you can reduce the fraction to a decimal number but you need to first write the correct fraction.

Explain what is wrong with each of the following diagrams.

The hypotenuse is too small. The hypotenuse is the longest side of the right-angled triangle and in this case one side of the triangle is given as being larger than the hypotenuse.

The hypotenuse is too small. The hypotenuse is the longest side of the right-angled triangle and in this case one side of the triangle is given as being larger than the hypotenuse.

5.4 Reciprocal ratios

Each of the three trigonometric ratios has a reciprocal. The reciprocals: cosecant (cosec), secant () and cotangent (), are defined as follows:

We can also define these reciprocals for any right-angled triangle:

Note that:

This video covers the three reciprocal ratios for , and .

Video: 2FNV

You may see cosecant abbreviated as .

5.5 Calculator skills

In this section we will look at using a calculator to determine the values of the trigonometric ratios for any angle. For example we might want to know what the value of is or what the value of is.

When doing calculations involving the reciprocal ratios you need to convert the reciprocal ratio to one of the standard trigonometric ratios: , and as this is the only way to calculate these ratios on your calculator.

Most scientific calculators are quite similar but these steps might differ depending on the calculator you use. Make sure your calculator is in “degrees” mode.

Note that . This also applies for the other trigonometric ratios.

Worked example 2: Using your calculator

Use your calculator to calculate the following (correct to decimal places):

The following shows the keys to press on a Casio calculator. Other calculators work in a similar way. On a Casio calculator ( is automatically added after pressing , and so you just need to press after typing in the angle to close the brackets.

Press

Press

Press

OR

Press

Press

OR

Press

Press

OR

Press

Press

OR

Press

First write in terms of : (since there is no “” button on your calculator).

Press

First write in terms of : (since there is no “” button on your calculator).

Press

Worked example 3: Calculator work using substitution

If and , use your calculator to determine whether the following statement is true or false:

Calculate the left hand side of the equation

Press

Write the final answer

LHS = RHS therefore the statement is true.

Exercise 5.2

Use your calculator to determine the value of the following (correct to decimal places):

If and , use a calculator to determine whether the following statements are true or false:

LHS:

RHS:

Therefore the statement is true.

LHS:

RHS:

Therefore the statement is false.

LHS:

RHS:

Therefore the statement is true.

LHS:

RHS:

Therefore the statement is false.

Solve for in .

To solve this problem we need to recall from exponents that if then . Then we note that . Now we can solve the problem:

5.6 Special angles

For most angles we need a calculator to calculate the values of , and . However, there are some angles we can easily work out the values for without a calculator as they produce simple ratios. The values of the trigonometric ratios for these special angles, as well as the triangles from which they are derived, are shown below.

Remember that the lengths of the sides of a right-angled triangle must obey the Theorem of Pythagoras: the square of the hypotenuse equals the sum of the squares of the two other sides.

30°

45°

60°

1

These values are useful when we need to solve a problem involving trigonometric ratios without using a calculator.

Exercise 5.3

Select the closest answer for each expression from the list provided:

Solve for in the following triangle, in surd form:

Solve for in the following triangle, in surd form:

Calculate the value of the following without using a calculator:

For both ratios the angle given is 45°. This is one of the special angles. We note that using special angles.

We are given angles of 45° and 60°. These are both special angles. We note that and using special angles.

For both ratios the angle given is 60°. This is one of the special angles. We note that and using special angles.

Evaluate the following without using a calculator. Select the closest answer from the list provided.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

We need to use special angles to help us solve this problem. First write down each ratio using special angles and then simplify the answer.

Use special angles to show that:

We are told to use special angles, so we first write each ratio using special angles and then simplify each side of the equation.

LHS:

RHS:

Therefore the equation is true.

We are told to use special angles, so we first write each ratio using special angles and then simplify each side of the equation.

LHS:

RHS

Therefore the equation is true.

We are told to use special angles, so we first write each ratio using special angles and then simplify each side of the equation.

LHS:

RHS:

Therefore the equation is true.

Use the definitions of the trigonometric ratios to answer the following questions:

Explain why for all values of .

The sine ratio is defined as . In any right-angled triangle, the hypotenuse is the side of longest length. Therefore the maximum length of the opposite side is equal to the length of the hypotenuse. The maximum value of the sine ratio is then .

Explain why has a maximum value of .

The cosine ratio is defined as . In any right-angled triangle, the hypotenuse is the side of longest length. Therefore the maximum length of the adjacent side is equal to the length of the hypotenuse. The maximum value of the cosine ratio is then .

Is there a maximum value for ?

The tangent ratio is defined as . Since the opposite and adjacent sides can have any value (so long as the length of the side is less than or equal to the length of the hypotenuse), there is no maximum value for the tangent ratio.

5.7 Solving trigonometric equations

In this section we will first look at finding unknown lengths in right-angled triangles and then we will look at finding unknown angles in right-angled triangles. Finally we will look at how to solve more general trigonometric equations.

Finding lengths

From the definitions of the trigonometric ratios and what we have learnt about determining the values of these ratios for any angle we can now use this to help us find unknown lengths in right-angled triangles. The following worked examples will show you how.

Worked example 4: Finding lengths

Find the length of in the following right-angled triangle using the appropriate trigonometric ratio (round your answer to two decimal places).

Identify the opposite and adjacent sides and the hypotenuse with reference to the given angle

Remember that the hypotenuse side is always opposite the right angle, it never changes position. The opposite side is opposite the angle we are interested in and the adjacent side is the remaining side.

Rearrange the equation to solve for

Use your calculator to find the answer

Worked example 5: Finding lengths

Find the length of in the following right-angled triangle using the appropriate trigonometric ratio (round your answer to two decimal places).

Identify the opposite and adjacent sides and the hypotenuse with reference to the given angle

Remember that the hypotenuse side is always opposite the right angle, it never changes position. The opposite side is opposite the angle we are interested in and the adjacent side is the remaining side.

Rearrange the equation to solve for

Use your calculator to find the answer

Worked example 6: Finding lengths

Find the length of and in the following right-angled triangle using the appropriate trigonometric ratio (round your answers to two decimal places).

Identify the opposite and adjacent sides and the hypotenuse with reference to the given angle

Rearrange the equations to solve for and

Use your calculator to find the answers

The following video shows an example of finding unknown lengths in a triangle using the trigonometric ratios.

Video: 2FQT

Exercise 5.4

In each triangle find the length of the side marked with a letter. Give your answers correct to decimal places.

Write down two ratios for each of the following in terms of the sides: .

We note that triangles and both contain angle so we can use these triangles to write down the ratios:

We note that triangles and both contain angle so we can use these triangles to write down the ratios:

We note that triangles and both contain angle so we can use these triangles to write down the ratios:

In , , and . Calculate and (correct to decimal places).

Sketch the triangle:

To find we use the sine ratio:

To find we can use the cosine ratio:

Therefore

Calculate and in the following diagram.

To find we use and the tangent ratio. To find we use and the tangent ratio.

Therefore .

Finding an angle

If the length of two sides of a triangle are known, the angles can be calculated using trigonometric ratios. In this section, we are finding angles inside right-angled triangles using the ratios of the sides.

Worked example 7: Finding angles

Find the value of in the following right-angled triangle using the appropriate trigonometric ratio.

Identify the opposite and adjacent sides with reference to the given angle and the hypotenuse

In this case you have the opposite side and the adjacent side for angle .

Use your calculator to solve for

To solve for , you will need to use the inverse tangent function on your calculator. This works backwards by using the ratio of the sides to determine the angle which resulted in that ratio.

Press

Write the final answer

Exercise 5.5

Determine in the following right-angled triangles:

We have now seen how to solve trigonometric equations in right-angled triangles. We can use the same techniques to help us solve trigonometric equations when the triangle is not shown.

Worked example 8: Solving trigonometric equations

Find the value of if .

Use your calculator to solve for

To solve for , you will need to use the inverse cosine function on your calculator. This works backwards by using the ratio of the sides to determine the angle which resulted in that ratio.

Press

Write the final answer

Worked example 9: Solving trigonometric equations

Find the value of if .

Rearrange the equation

We need to rearrange the equation so that is on one side of the equation.

Use your calculator to solve for

To solve for , you will need to use the inverse sine function on your calculator. This works backwards by using the ratio of the sides to determine the angle which resulted in that ratio.

Press .

Write the final answer

When you are solving trigonometric equations you might find that you get an error when you try to calculate or (remember that both the sine and cosine functions have a maximum value of 1). For these cases there is no solution to the equation.

If learners get a math error on their calculator encourage them to think about what might have happened. It is also important to ensure that they know they must write down no solution rather than math error when this happens.

Worked example 10: Solving trigonometric equations

Solve for : .

Convert to

There is no “” button on the calculator and so we need to convert to so we can find .

Rearrange the equation

We need to rearrange the equation so that we have on one side of the equation.

Use your calculator to solve for

To solve for , you will need to use the inverse cosine function on your calculator. This works backwards by using the ratio of the sides to determine the angle which resulted in that ratio.

Press math error

In this case we get an error when we try to do the calculation. This is because is greater than 1 and the maximum value of the cosine function is 1. Therefore there is no solution. It is important in this case to write no solution and not math error.

Write the final answer

There is no solution.

Exercise 5.6

Determine the angle (correct to decimal place):

If and , use your calculator to evaluate each of the following, correct to 3 decimal places.

In each of the following find the value of correct to two decimal places.

5.8 Defining ratios in the Cartesian plane

We have defined the trigonometric ratios using right-angled triangles. We can extend these definitions to any angle, noting that the definitions do not rely on the lengths of the sides of the triangle, but on the size of the angle only. So if we plot any point on the Cartesian plane and then draw a line from the origin to that point, we can work out the angle between the -axis and that line. We will first look at this for two specific points and then look at the more general case.

Finding an angle for specific points

In the figure below points and have been plotted. A line from the origin () to each point is drawn. The dotted lines show how we can construct right-angled triangles for each point. The dotted line must always be drawn to the -axis. Now we can find the angles and :

We can also extend the definitions of the reciprocals in the same way.

From the coordinates of , we can see that and . Therefore, we know the length of the side opposite is and the length of the adjacent side is . Using: we calculate that .

We can also use the theorem of Pythagoras to calculate the hypotenuse of the triangle and then calculate using:

Consider point . We define as the angle formed between line and the positive -axis. This is called the standard position of an angle. Angles are always measured from the positive -axis in an anti-clockwise direction. Let be the angle formed between the line and the negative -axis such that .

From the coordinates of , we know the length of the side opposite is and the length of the adjacent side is . Using: we calculate that . Therefore .

Similarly, an alternative method is to calculate the hypotenuse using the theorem of Pythagoras and calculate using:

Finding any angle

If we were to draw a circle centred on the origin () and passing through the point , then the length from the origin to point is the radius of the circle, which we denote . We denote the angle formed between the line and the -axis as .

We can rewrite all the trigonometric ratios in terms of , and . The general definitions for the trigonometric ratios are:

The CAST diagram

The Cartesian plane is divided into quadrants in an anti-clockwise direction as shown in the diagram below. Notice that is always positive but the values of and change depending on the position of the point in the Cartesian plane. As a result, the trigonometric ratios can be positive or negative. The letters C, A, S and T indicate which of the ratios are positive in each quadrant:

This diagram is known as the CAST diagram.

We note the following using the general definitions of the trigonometric ratios:

Quadrant I

Both the and values are positive so all ratios are positive in this quadrant.

Quadrant II

The values are positive therefore and cosec are positive in this quadrant (recall that and cosec are defined in terms of and ).

Quadrant III

Both the and the values are negative therefore and are positive in this quadrant (recall that and are defined in terms of and ).

Quadrant IV

The values are positive therefore and are positive in this quadrant (recall that and are defined in terms of and ).

The hypotenuse, , is a length, and is therefore always positive.

The following video provides a summary of the trigonometric ratios in the Cartesian plane.

Video: 2FSN

Special angles in the Cartesian plane

When working in the Cartesian plane we include two other special angles in right-angled triangles: ° and °.

Notice that when the length of the opposite side is equal to and the length of the adjacent side is equal to the length of the hypotenuse. Therefore:

When the length of the adjacent side is equal to and the length of the opposite side is equal to the length of the hypotenuse. Therefore:

Now we can extend our knowledge of special angles.

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Worked example 11: Ratios in the Cartesian plane

is a point on the Cartesian plane with origin . is the angle between and the positive -axis. Without using a calculator, determine the value of:

Sketch point in the Cartesian plane and label the angle

Use the theorem of Pythagoras to calculate

Note is positive as it is the radius of the circle.

Substitute values for , and into the required ratios

We note that , and .

Worked example 12: Ratios in the Cartesian plane

is an angle in the third quadrant where is a point on the positive -axis and is the point . is .

Determine, without using a calculator, the value of .

Prove that without using a calculator.

Sketch point in the Cartesian plane and label the angle

Use the theorem of Pythagoras to calculate

Given that lies in the third quadrant, must be negative.

Substitute values for , and and simplify

, and .

LHS

RHS

Therefore the LHS = RHS and we have proved that .

Whenever you have to solve trigonometric problems without a calculator, it can be very helpful to make a sketch.

Exercise 5.7

is a point in the Cartesian plane. Determine without using a calculator:

is the hypotenuse of . We can calculate the length of using the theorem of Pythagoras:

From the diagram and the first question we know that , and .

From the diagram and the first question we know that , and .

From the diagram and the first question we know that , and .

If and is an obtuse angle, determine:

We first need to determine , and .

Therefore and .

We are told that the angle is obtuse. An obtuse angle is greater than 90° but less than 180°. Therefore the angle is in the second quadrant and is negative. Therefore .

Next draw a sketch:

Now we can determine :

From the first question we have a sketch of the angle and , and .

Given , where . Determine the following in terms of :

We first need to determine , and . We are given and so we can use this to find and .

Therefore and .

We are told that . Therefore the angle is in the first quadrant. Even though we do not know the value of we can draw a rough sketch:

Now we can determine :

From the first question we have a sketch of the angle and , and .

From the first question we have a sketch of the angle and , and .

From the first question we have a sketch of the angle and , and .

Given: and . Determine the value of:

We are given an equation with in it. We can therefore rearrange this equation to find :

We first need to determine , and . In the first question we found that and so we can use this to find and .

Therefore and .

We are told that . Therefore the angle is in the third quadrant and . We can draw a rough sketch of the angle:

We can now find :

If and find the following, without the use of a calculator:

We first need to determine , and . We are given and so we can use this to find and .

Therefore and (remember that cannot be negative).

We are told that . Therefore the angle is in either the second or the third quadrant. From the value of we see that the angle must lie in the third quadrant and .

Now we can determine :

From the first part we have , and so we can find .

From the first part we have , and so we can find .

Find the value of without using a calculator, given that , where .

We first need to determine , and . We are given and so we can use this to find and .

Therefore and .

We are told that . Therefore the angle is in either the second or the third quadrant. From the value of we see that the angle must lie in the second quadrant.

Now we can determine :

If and determine the following with the aid of a diagram (not a calculator):

We first need to determine , and . We are given and so we can use this to find and .

Therefore and .

We are told that . Therefore the angle is in either the first or the third quadrant. From the value of we see that the angle must lie in the third quadrant.

Now we can determine :

From the first part we have that , and .

is a point with co-ordinates on a Cartesian plane. forms an angle, , with the positive -axis. Set up a diagram and use it to answer the following questions.

Find the distance .

We are given . Therefore the angle lies in the first quadrant. We can sketch this and use our sketch to find , and .

Therefore and . We can calculate using the theorem of Pythagoras. From the diagram we note that .

From the previous question we have that , and .

From the previous question we have that , and .

From the previous question we have that , and .

From the previous question we have that , and .

From the previous question we have that , and .

From the previous question we have that , and .

From the previous question we have that , and .

Given the following diagram and that .

State two sets of possible values of and .

We first need to use the given information to find a possible set of values for and .

Using and the fact that we can determine that and . Now we can find :

From the diagram we see that must be negative.

This gives us one possible set of values for and : and .

Now we note that we can simply double the size of the circle and the trigonometric ratios will stay the same. We could even multiply the radius of the circle by any integer and the trigonometric ratios will still remain the same.

Therefore the possible sets of values for are multiples of . Two possible sets are and .

If , state the values of and .

First note that in the original diagram . Now we are multiplying by 4. This also means that the and values must be multiplied by 4.

Therefore and .

Hence determine without the use of a calculator the value of .

The question states: “hence”. This means we must use the scaled values for and not the original values. We know that , and .

Notice how the answer reduced to the original values of and as we would expect from the first question.

If and , determine without the use of a calculator the value of

We first need to determine , and . We are given and so we can use this to find and .

Therefore and .

We are told that . Therefore the angle is in either the first or the second quadrant. From the values of and we see that the angle must lie in the second quadrant.

Now we can determine :

5.9 Chapter summary

Presentation: 2FT2

We can define three trigonometric ratios for right-angled triangles: sine (), cosine () and tangent ()

These ratios can be defined as:

Each of these ratios have a reciprocal: cosecant (), secant () and cotangent ().

These ratios can be defined as:

We can use the principles of solving equations and the trigonometric ratios to help us solve simple trigonometric equations.

For some special angles (0°, 30°, 45°, 60° and 90°), we can easily find the values of , and without using a calculator.

We can extend the definitions of the trigonometric ratios to any angle.

Exercise 5.8

State whether each of the following trigonometric ratios has been written correctly.

We recall the definition of the sine ratio: . Therefore this trigonometric ratio has not been written correctly.

We recall the definition of the tangent ratio: . Therefore this trigonometric ratio has been written correctly.

We recall the definition of the secant ratio: . Therefore this trigonometric ratio has not been written correctly.

Use your calculator to evaluate the following expressions to two decimal places:

Use the triangle below to complete the following:

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Use the triangle below to complete the following:

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Remember to first identify the hypotenuse, opposite and adjacent sides for the given angle. Then write down the correct fraction for each ratio. You can confirm your answer by using your calculator to find the value of the ratio for that angle.

Evaluate the following without using a calculator. Select the closest answer from the list provided.

Without using a calculator, determine the value of:

These are all special angles.

Solve for in the following triangle, in surd form:

Solve for in the following triangle, in surd form:

A right-angled triangle has hypotenuse . Find the length of the other two sides if one of the angles of the triangle is °.

First draw a diagram:

Next we get:

Now we can use the theorem of Pythagoras to find the other side:

Therefore the other two sides are and .

Solve for to the nearest integer.

Calculate the unknown lengths in the diagrams below:

For all of these we use the appropriate trigonometric ratio or the theorem of Pythagoras to solve.

To find and we use :

To find we use :

To find we use

Next we use the theorem of Pythagoras to find the third side, so we can use trig functions to find :

We use to find :

Next we use the theorem of Pythagoras to find the third side, so we can use trig functions to find and :

We use to find :

And finally we find using the theorem of Pythagoras:

The final answers are: , , , , , and .

In , , and . The perpendicular line from to intersects at . Calculate:

the length

First draw a sketch:

Since we are told that we can use to find .

the length

We can use to find .

the angle

We know the length of and we have found the length of , so we can work out the length of :

Since we know two sides and an angle we can use to find the angle:

In the following triangle find the size of .

We use to find :

Next we find :

And then we use to find the angle:

In the following triangle find the length of side :

We use the angles in a triangle to find :

Then we find :

Now we can use to find :

Then we find also using :

Finally we can find :

Determine

The length of

We note that and , therefore . So we need to find :

The value of

Given that . Calculate .

We first calculate by using the given information about and .

Next we find :

Now we can calculate :

Finally we can calculate :

Solve for if is a positive, acute angle:

First we note that:

Now we can solve for :

If , and , use your calculator to evaluate each of the following, correct to 2 decimal places.

If and , use a sketch to determine:

Find , and

Therefore and .

Draw a sketch:

Now we can find :

We have , and from the first question.

Given and , find the angle between the line through and and the -axis.

First draw a diagram:

Next we note that the distance from to the -axis is ( is units up from the -axis) and that the distance from to is .

We use the tangent ratio to find the angle:

Therefore the angle between line and the -axis is °.

Given and , find the angle between the line through and and the -axis.

First draw a diagram:

Next we note that the distance from to the -axis is (although is the distance is positive). The distance from to the point where the perpendicular line from intercepts the -axis is .

We use the tangent ratio to find the angle:

Therefore the angle between line and the -axis is °.

Given the points , and . Find the angle .

First draw a sketch:

To find we look at and in turn. These two triangles will each give one part of the angle that we want.

In triangle we can use the tangent ratio. is units and is unit.

In triangle we also use the tangent ratio. is units and is units.

Now we add these two angles together to get the angle we want to find:

A triangle with angles °, ° and ° has a perimeter of . Find the length of each side of the triangle.

First draw a sketch:

We construct a perpendicular bisector and now we have a right-angled triangle to work with. We can use either of these two triangles.

We know . Rearranging gives: . We can use the cos ratio to find :

From the perimeter we get:

Therefore the lengths of the sides are , and .

Determine the area of .

Chapter 6: Functions

6.1 Introduction

This chapter covers the concept of a function and representing functions using tables, graphs, words and formulae. Straight line graphs were covered in grade 9 and are revised here. Parabolas, hyperbolas and exponential graphs are introduced here. Graphs for sine, cosine and tangent functions are also introduced here.

A more formal definition of a function is only covered in grade 12. At this level learners should know the terms independent (input) and dependent (output) variables as well as how these vary.

Summaries should be compiled for each type of graph and include the effects of a (vertical stretch and/or reflection in x) and q (vertical shift).

Remember that graphs in some practical applications might be discrete or continuous.

Encourage learners to state restrictions, particularly for quadratic functions.

Learners must understand that has no real solutions for .

Sketching graphs is based on knowing the effects of and and using these to determine the shape of the graph.

A tool such as this one can be used to plot graphs for classroom use. If you use this tool for plotting trigonometric graphs the values on the -axis will not be in degrees.

Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can take input from many variables, but always give the same output, unique to that function.

Functions also allow us to visualise relationships in terms of graphs, which are much easier to read and interpret than lists of numbers.

Figure 6.1: A cricket player facing a delivery. If a cricket player is hit on his batting pads and the umpire thinks that the ball would have hit the stumps behind him, he is given out LBW (leg before wicket). At professional levels of the game, sophisticated software is used to determine if the ball will hit the stumps. The software uses functions to predict the flight of the ball if the cricket players leg had not been in the way.

Some examples of functions include:

Money as a function of time. You never have more than one amount of money at any time because you can always add everything to give one total amount. By understanding how your money changes over time, you can plan to spend your money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much.

Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, where you are and many more. But the important thing is that there is only one temperature output when you measure it in a specific place.

Location as a function of time. You can never be in two places at the same time. If you were to plot the graphs of where two people are as a function of time, the place where the lines cross means that the two people meet each other at that time. This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.

Function

A function is a mathematical relationship between two variables, where every input variable has one output variable.

Dependent and independent variables

In functions, the -variable is known as the input or independent variable, because its value can be chosen freely. The calculated -variable is known as the output or dependent variable, because its value depends on the chosen input value.

Set notation

Examples:

The set of all -values such that is an element of the set of real numbers and is greater than .

The set of all -values such that is a natural number, is greater than and is less than or equal to .

The set of all -values such that is an integer and is less than or equal to .

Interval notation

It is important to note that this notation can only be used to represent an interval of real numbers.

Examples:

Round brackets indicate that the number is not included. This interval includes all real numbers greater than but not equal to and less than but not equal to .

Round brackets are always used for positive and negative infinity. This interval includes all real numbers less than, but not equal to .

A square bracket indicates that the number is included. This interval includes all real numbers greater than or equal to and less than but not equal to .

Function notation

This is a very useful way to express a function. Another way of writing is . We say “ of is equal to ”. Any letter can be used, for example, , , , etc.

Determine the output value:

“Find the value of the function for ” can be written as: “find ”.

Replace with :

This means that when , the value of the function is .

Determine the input value:

“Find the value of that will give a -value of ” can be written as: “find if ”.

We write the following equation and solve for :

This means that when the value of the function is .

Representations of functions

Functions can be expressed in many different ways for different purposes.

Words: “The relationship between two variables is such that one is always less than the other.”

Mapping diagram:

Table:

Input variable

Output variable

Set of ordered number pairs: , ,

Algebraic formula:

Graph:

Domain and range

The domain of a function is the set of all independent -values from which the function produces a single -value for each -value.

The range is the set of all dependent -values which can be obtained using an independent -value.

Exercise 6.1

Write the following in set notation:

Write the following in interval notation:

Complete the following tables and identify the function.

Plot the following points on a graph.

Note that this graph is scaled. Each value for and has been multiplied by . This process does not change the function, but it stretches the graph, thereby making it easier to read.

Create a table of values from the function given and then plot the function. Your table must have at least 5 ordered pairs.

If the functions are given, find the value of the following:

Regardless of the value of , the output is always 3.

Regardless of the value of , the output is always 3.

The cost of petrol and diesel per litre are given by the functions and , where:

Use this information to answer the following:

Evaluate

Evaluate

How many litres of petrol can you buy with ?

How many litres of petrol can you buy with ?

How much more expensive is petrol than diesel? Show you answer as a function.

A ball is rolling down a slope. The graph below shows the relationship between the distance and the time.

Use this information to answer the following:

After how much further does the ball have to roll?

What is the range of the function?

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What is the domain of the function, and what does it represent?

The domain is . It represents the total time taken to reach the bottom of the slope.

James and Themba both throw a stone from the top of a building into a river. The path travelled by the stones can be described by quadratic equations. describes the path of the stone thrown by James and describes the path of Themba's stone.

What is the height of the building that they stood on?

Both functions have a maximum value of . This can be found by letting in each of the two functions and is represented by point A on the graph above.

How far did James throw his stone before it hit the river surface?

James threw his stone before it hit the river surface.

How much farther did Themba throw his stone before it hit the river surface?

Themba threw his stone before it hit the river surface.

Therefore Themba threw his stone farther than James did.

6.2 Linear functions

Functions of the form

Functions of the form are called straight line functions. In the equation, , and are constants and have different effects on the graph of the function.

Worked example 1: Plotting a straight line graph

Complete the following table for and plot the points on a set of axes.

Join the points with a straight line.

Determine the domain and range.

About which line is symmetrical?

Using the graph, determine the value of for which . Confirm your answer graphically.

Where does the graph cut the axes?

Substitute values into the equation

Plot the points and join with a straight line curve

From the table, we get the following points and the graph:

Determine the domain and range

Domain:

Range:

Determine the value of for which

From the graph we see that when , . This gives the point .

Determine the intercept

The function intercepts the axes at the origin .

Functions of the form

The effects of and on a straight line graph

On the same set of axes, plot the following graphs:

Use your results to deduce the effect of different values of on the graph.

On the same set of axes, plot the following graphs:

Use your results to deduce the effect of different values of on the graph.

The effect of

We notice that the value of affects the slope of the graph. As increases, the gradient of the graph increases.

If then the graph increases from left to right (slopes upwards).

If then the graph increases from right to left (slopes downwards). For this reason, is referred to as the gradient of a straight-line graph.

The effect of

We also notice that the value of affects where the graph cuts the -axis. For this reason, is known as the -intercept.

If the graph shifts vertically upwards.

If the graph shifts vertically downwards.

The effect of and on a straight line graph.

You can use this Phet simulation to help you see the effects of changing and .

Discovering the characteristics

The standard form of a straight line graph is the equation .

Domain and range

The domain is because there is no value of for which is undefined.

The range of is also because can take on any real value.

Intercepts

The -intercept:

Every point on the -axis has an -coordinate of . Therefore to calculate the -intercept, let .

For example, the -intercept of is given by setting :

This gives the point .

The -intercept:

Every point on the -axis has a -coordinate of . Therefore to calculate the -intercept, let .

For example, the -intercept of is given by setting :

This gives the point .

Sketching graphs of the form

In order to sketch graphs of the form, , we need to determine three characteristics:

sign of

-intercept

-intercept

Dual intercept method

Only two points are needed to plot a straight line graph. The easiest points to use are the -intercept and the -intercept.

Worked example 2: Sketching a straight line graph using the dual intercept method

Sketch the graph of using the dual intercept method.

Examine the standard form of the equation

. This means that the graph increases as increases.

Calculate the intercepts

For the -intercept, let ; therefore . This gives the point .

For the -intercept, let ; therefore . This gives the point .

Plot the points and draw the graph

Gradient and -intercept method

We can draw a straight line graph of the form using the gradient () and the -intercept ().

We calculate the -intercept by letting . This gives us one point for drawing the graph and we use the gradient to calculate the second point.

The gradient of a line is the measure of steepness. Steepness is determined by the ratio of vertical change to horizontal change:

For example, , therefore and the graph slopes upwards.

Worked example 3: Sketching a straight line graph using the gradient–intercept method

Sketch the graph of using the gradient-intercept method.

Use the intercept

, which gives the point .

Use the gradient

Start at . Move unit up and units to the right. This gives the second point .

Or start at , move unit down and units to the left. This gives the second point .

Plot the points and draw the graph

Always write the function in the form and take note of . After plotting the graph, make sure that the graph increases if and that the graph decreases if .

Exercise 6.2

Determine the -intercept and the -intercept of the following equations.

-intercept and -intercept

-intercept and -intercept

-intercept and -intercept

In the graph below there is a function with the equation . Determine the values of (the gradient of the line) and (the -intercept of the line).

To determine , we use the coordinates of any other point on the line apart from the one used for the -intercept. In this solution, we have chosen the coordinates of point which are .

From the -intercept .

and .

The graph below shows a function with the equation . Determine the values of (the gradient of the line) and (the -intercept of the line).

To determine , we use the coordinates of any other point on the line apart from the one used for the -intercept. In this solution, we have chosen the coordinates of point which are .

From the -intercept .

, and .

List the and -intercepts for the following straight line graphs. Indicate whether the graph is increasing or decreasing:

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is increasing ().

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is increasing ().

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is increasing ().

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is decreasing ().

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is increasing ().

To find the -intercept we set and to find the -intercept we set . This gives the point . The graph is horizontal.

To find the -intercept we set and to find the -intercept we set . This gives the same point for both intercepts: . The graph is increasing ().

To find the -intercept we set and to find the -intercept we set . This gives the points and . The graph is increasing ().

State whether the following are true or not.

The gradient of is .

False

Therefore the gradient is .

The -intercept of is .

True

The gradient of is .

True

The gradient of is .

False

The -intercept of is .

False

Write the following in standard form ():

Look at the graphs below. Each graph is labelled with a letter. In the questions that follow, match any given equation with the label of a corresponding graph.

E

A

B

F

D

C

For the functions in the diagram below, give the equation of each line:

The -intercept is , therefore .

Therefore

The -intercept is , therefore .

Therefore

The -intercept is , therefore .

Therefore

The -intercept is , therefore .

Therefore

Sketch the following functions on the same set of axes, using the dual intercept method. Clearly indicate the coordinates of the intercepts with the axes and the point of intersection of the two graphs: and .

For :

We first write the equation in standard form: . From this we see that the -intercept is . The -intercept is .

For :

We first write the equation in standard form: . From this we see that the -intercept is . The -intercept is .

To find the point of intersection we need to solve the two equations simultaneously. We can use the standard form of the first equation and substitute this value of into the second equation:

Substitute the value of back into the first equation:

Therefore the graphs intersect at .

Now we can sketch the graphs:

On the same set of axes, draw the graphs of and using the gradient-intercept method.

For the -intercept is 3. The gradient is .

To get the second point we start at and move 3 units up and 1 unit to the left. This gives the second point . Or we can move 3 units down and 1 unit right to get .

For the -intercept is 1. The gradient is .

To get the second point we start at and move 1 unit up and 3 units to the right. This gives the second point . Or we can move 1 unit down and 3 units left to get .

Now we can sketch the graphs.

6.3 Quadratic functions

Functions of the form

Functions of the general form are called parabolic functions. In the equation , and are constants and have different effects on the parabola.

Worked example 4: Plotting a quadratic function

Complete the following table for and plot the points on a system of axes.

Join the points with a smooth curve.

The domain of is . Determine the range.

About which line is symmetrical?

Determine the value of for which . Confirm your answer graphically.

Where does the graph cut the axes?

Substitute values into the equation

Plot the points and join with a smooth curve

From the table, we get the following points:

Determine the domain and range

Domain:

From the graph we see that for all values of , .

Range:

Find the axis of symmetry

is symmetrical about the -axis. Therefore the axis of symmetry of is the line .

Determine the -value for which

See points and on the graph.

Determine the intercept

The function intercepts the axes at the origin .

We notice that as the value of increases from to , decreases.

At the turning point , .

As the value of increases from to , increases.

Functions of the form

The effects of and on a parabola.

Complete the table and plot the following graphs on the same system of axes:

Use your results to deduce the effect of .

Complete the table and plot the following graphs on the same system of axes:

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).

For , the graph of is shifted vertically upwards by units. The turning point of is above the -axis.

For , the graph of is shifted vertically downwards by units. The turning point of is below the -axis.

The effect of

The sign of determines the shape of the graph.

For , the graph of is a “smile” and has a minimum turning point at . The graph of is stretched vertically upwards; as gets larger, the graph gets narrower.

For , as gets closer to , the graph of gets wider.

For , the graph of is a “frown” and has a maximum turning point at . The graph of is stretched vertically downwards; as gets smaller, the graph gets narrower.

For , as gets closer to , the graph of gets wider.

The effect of and on a parabola.

You can use this Phet simulation to help you see the effects of changing and for a parabola.

Discovering the characteristics

The standard form of the equation of a parabola is .

Domain and range

The domain is because there is no value for which is undefined.

If then we have:

Therefore if , the range is . Similarly, if then the range is .

Worked example 5: Domain and range of a parabola

If , determine the domain and range of the function.

Determine the domain

The domain is because there is no value for which is undefined.

Determine the range

The range of can be calculated as follows:

Therefore the range is .

Intercepts

The -intercept:

Every point on the -axis has an -coordinate of , therefore to calculate the -intercept let .

For example, the -intercept of is given by setting :

This gives the point .

The -intercepts:

Every point on the -axis has a -coordinate of , therefore to calculate the -intercept let .

For example, the -intercepts of are given by setting :

There is no real solution, therefore the graph of does not have -intercepts.

Turning points

The turning point of the function of the form is determined by examining the range of the function.

If , the graph of is a “smile” and has a minimum turning point at .

If , the graph of is a “frown” and has a maximum turning point at .

Axes of symmetry

The axis of symmetry for functions of the form is the -axis, which is the line .

Sketching graphs of the form

In order to sketch graphs of the form , we need to determine the following characteristics:

sign of

-intercept

-intercept

turning point

Worked example 6: Sketching a parabola

Sketch the graph of . Mark the intercepts and turning point.

Examine the standard form of the equation

We notice that . Therefore the graph is a “smile” and has a minimum turning point.

Calculate the intercepts

For the -intercept, let :

This gives the point .

For the -intercepts, let :

This gives the points and .

Determine the turning point

From the standard form of the equation we see that the turning point is .

Plot the points and sketch the graph

Domain:

Range:

The axis of symmetry is the line .

Worked example 7: Sketching a parabola

Sketch the graph of . Mark the intercepts and the turning point.

Examine the standard form of the equation

We notice that . Therefore the graph is a “frown” and has a maximum turning point.

Calculate the intercepts

For the -intercept, let :

This gives the point .

For the -intercepts let :

There is no real solution, therefore there are no -intercepts.

Determine the turning point

From the standard form of the equation we see that the turning point is .

Plot the points and sketch the graph

Domain: .

Range: .

The axis of symmetry is the line .

Exercise 6.3

The graph below shows a quadratic function with the following form: . Two points on the parabola are shown: Point A, the turning point of the parabola, at , and Point B is at . Calculate the values of and .

The value of is 4.

The graph below shows a quadratic function with the following form: . Two points on the parabola are shown: Point A, the turning point of the parabola, at , and Point B is at . Calculate the values of and .

The value of is -3.

Given the following equation:

Calculate the -coordinate of the -intercept.

The -coordinate of the -intercept is .

Now calculate the -intercepts. Your answer must be correct to 2 decimal places.

The -intercepts are and .

Given the following equation:

Calculate the -coordinate of the -intercept.

The -coordinate of the -intercept is .

Now calculate the -intercepts. Your answer must be correct to 2 decimal places.

The -intercepts are and .

Given the following graph, identify a function that matches each of the following equations:

Given the following graph, identify a function that matches each of the following equations:

Two parabolas are drawn: and .

Find the values of and .

is the -intercept of the function , therefore

To find we use one of the points on the graph (e.g. ):

;

Find the values of and .

is the -intercept of the function , therefore

To find , we use one of the points on the graph (e.g. ):

;

Find the values of for which .

These are the points where lies above .

From the graph we see that lies above when: or

For what values of is increasing?

increases from the turning point , i.e. for .

Show that if the range of is .

Because the square of any number is always positive we get: .

If we multiply by where then the sign of the inequality is reversed:

Adding to both sides gives

And so

This gives the range as .

Draw the graph of the function showing all intercepts with the axes.

The -intercept is . The -intercepts are given by setting :

Therefore the -intercepts are: and .

Now we can sketch the graph:

6.4 Hyperbolic functions

Functions of the form

Functions of the general form are called hyperbolic functions.

Worked example 8: Plotting a hyperbolic function

Complete the following table for and plot the points on a system of axes.

0

Join the points with smooth curves.

What happens if ?

Explain why the graph consists of two separate curves.

What happens to as the value of becomes very small or very large?

The domain of is . Determine the range.

About which two lines is the graph symmetrical?

Substitute values into the equation

undefined

Plot the points and join with two smooth curves

From the table we get the following points: , , , , , , , , ,

For the function is undefined. This is called a discontinuity at .

therefore we can write that . Since the product of two positive numbers and the product of two negative numbers can be equal to , the graph lies in the first and third quadrants.

Determine the asymptotes

As the value of gets larger, the value of gets closer to, but does not equal . This is a horizontal asymptote, the line . The same happens in the third quadrant; as gets smaller also approaches the negative -axis asymptotically.

We also notice that there is a vertical asymptote, the line ; as gets closer to , approaches the -axis asymptotically.

Determine the range

Domain:

From the graph, we see that is defined for all values except .

Range:

Determine the lines of symmetry

The graph of has two axes of symmetry: the lines and . About these two lines, one half of the hyperbola is a mirror image of the other half.

Functions of the form

The effects of and on a hyperbola.

On the same set of axes, plot the following graphs:

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs:

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).

For , the graph of is shifted vertically upwards by units.

For , the graph of is shifted vertically downwards by units.

The horizontal asymptote is the line and the vertical asymptote is always the -axis, the line .

The effect of

The sign of determines the shape of the graph.

If , the graph of lies in the first and third quadrants.

For , the graph of will be further away from the axes than .

For , as tends to , the graph moves closer to the axes than .

If , the graph of lies in the second and fourth quadrants.

For , the graph of will be further away from the axes than .

For , as tends to , the graph moves closer to the axes than .

The effects of and on a hyperbola.

Discovering the characteristics

The standard form of a hyperbola is the equation .

Domain and range

For , the function is undefined for . The domain is therefore .

We see that can be rewritten as:

This shows that the function is undefined only at .

Therefore the range is

Worked example 9: Domain and range of a hyperbola

If , determine the domain and range of the function.

Determine the domain

The domain is because is undefined only at .

Determine the range

We see that is undefined only at . Therefore the range is

Intercepts

The -intercept:

Every point on the -axis has an -coordinate of , therefore to calculate the -intercept let .

For example, the -intercept of is given by setting :

which is undefined, therefore there is no -intercept.

The -intercept:

Every point on the -axis has a -coordinate of , therefore to calculate the -intercept, let .

For example, the -intercept of is given by setting :

This gives the point .

Asymptotes

There are two asymptotes for functions of the form .

The horizontal asymptote is the line and the vertical asymptote is always the -axis, the line .

Axes of symmetry

There are two lines about which a hyperbola is symmetrical: and .

Sketching graphs of the form

In order to sketch graphs of functions of the form, , we need to determine four characteristics:

sign of

-intercept

-intercept

asymptotes

Worked example 10: Sketching a hyperbola

Sketch the graph of . Mark the intercepts and the asymptotes.

Examine the standard form of the equation

We notice that therefore the graph of lies in the first and third quadrant.

Calculate the intercepts

For the -intercept, let :

This is undefined, therefore there is no -intercept.

For the -intercept, let :

This gives the point .

Determine the asymptotes

The horizontal asymptote is the line . The vertical asymptote is the line .

Sketch the graph

Domain:

Range:

Worked example 11: Sketching a hyperbola

Sketch the graph of .

Examine the standard form of the equation

We see that therefore the graph lies in the second and fourth quadrants.

Calculate the intercepts

For the -intercept, let :

This is undefined, therefore there is no -intercept.

For the -intercept, let :

This gives the point .

Determine the asymptotes

The horizontal asymptote is the line . The vertical asymptote is the line .

Sketch the graph

Domain:

Range:

Axis of symmetry: and

Exercise 6.4

The following graph shows a hyperbolic equation of the form . Point A is shown at . Calculate the values of and .

Therefore and

The equation is .

The following graph shows a hyperbolic equation of the form . Point A is shown at . Calculate the values of and .

Therefore and

The equation is .

Given the following equation:

Determine the location of the -intercept.

There is no -intercept.

Determine the location of the -intercept. Give your answer as a fraction.

Given the following equation:

Determine the location of the -intercept.

There is no -intercept.

Determine the location of the -intercept.

Given the following graph, identify a function that matches each of the following equations:

Given the function: .

Draw the graph.

is negative and so the function lies in the second and fourth quadrants.

There is no -intercept or -intercept.

Instead we can plot the graph from a table of values.

Now we can plot the graph:

Does the point lie on the graph? Give a reason for your answer.

If we substitute the point into each side of the equation we get:

This satisfies the equation therefore the point does lie on the graph.

If the -value of a point on the graph is what is the corresponding -value?

Substitute in the value of :

What happens to the -values as the -values become very large?

The -values decrease as the -values become very large. The larger the denominator (), the smaller the result of the fraction ().

Give the equation of the asymptotes.

The graph is not vertically or horizontally shifted, therefore the asymptotes are and .

With the line as a line of symmetry, what is the point symmetrical to ?

Across the line of symmetry , the point symmetrical to is .

Given the function: .

Draw the graph.

is positive and so the function lies in the first and third quadrants.

There is no -intercept and no -intercept.

Instead we can plot the graph from a table of values.

How would the graph of compare with that of ? Explain your answer fully.

The graph is the graph of , vertically shifted upwards by units. They would be the same shape but the asymptote of would be , instead of (for ) and the axis of symmetry would be instead of .

Draw the graph of on the same set of axes, showing asymptotes, axes of symmetry and the coordinates of one point on the graph.

is positive and so the function lies in the first and third quadrants.

For there is no -intercept. The -intercept is at .

We can plot the graph from a table of values.

Sketch the functions given and describe the transformation performed on the first function to obtain the second function. Show all asymptotes.

and

is positive for both graphs and so both graphs lie in the first and third quadrants.

For both graphs there is no -intercept or -intercept.

Instead we can plot the graph from a table of values.

:

:

The asymptotes are and .

Now we can plot the graphs:

Magnification by 3

and

is positive for both graphs and so both graphs lie in the first and third quadrants.

For both graphs there is no -intercept. For there is no -intercept. For the -intercept is .

We can plot the graphs from a table of values.

:

:

The asymptotes for are and .

The asymptotes for are and .

Now we can plot the graphs:

Translation along the -axis by -1

and

:

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

We can plot the graph from a table of values.

The asymptotes are and .

:

is negative and so the graph lies in the second and fourth quadrants.

There is no -intercept and no -intercept.

We can plot the graph from a table of values.

The asymptotes are and .

Now we can plot the graphs:

Reflection on the -axis or reflection on the -axis.

and

is positive for both graphs and so both graphs lie in the first and third quadrants.

For both graphs there is no -intercept and no -intercept.

We can plot the graphs from a table of values.

:

:

The asymptotes for both graphs are and .

Now we can plot the graphs:

Reduction by 2

6.5 Exponential functions

Functions of the form

Functions of the general form are called exponential functions. In the equation and are constants and have different effects on the function.

Worked example 12: Plotting an exponential function

Complete the following table for each of the functions and draw the graphs on the same system of axes: , , .

At what point do these graphs intersect?

Explain why they do not cut the -axis.

Give the domain and range of .

As increases, does increase or decrease?

Which of these graphs increases at the slowest rate?

For and , the greater the value of , the steeper the curve of the graph. True or false?

Complete the following table for each of the functions and draw the graphs on the same system of axes: , ,

Give the -intercept for each function.

Describe the relationship between the graphs and .

Describe the relationship between the graphs and .

Give the domain and range of .

For and , the greater the value of , the steeper the curve of the graph. True or false?

Give the equation of the asymptote for the functions.

Substitute values into the equations

Plot the points and join with a smooth curve

We notice that all graphs pass through the point . Any number with exponent is equal to .

The graphs do not cut the -axis because you can never get by raising any non-zero number to the power of any other number.

Domain:

Range:

As increases, increases.

increases at the slowest rate because it has the smallest base.

True: the greater the value of , the steeper the graph of .

The -intercept is the point for all graphs. For any real number : .

is the reflection of about the -axis.

is the reflection of about the -axis.

Domain:

Range:

True: the greater the value of , the steeper the graph of .

The equation of the horizontal asymptote is , the -axis.

Functions of the form

CAPS states to only investigate the effects of and on an exponential graph. However it is also important for learners to see that has a different effect on the graph depending on if or .

For this reason the effect of is included in the investigation so that learners can see what happens when and when .

Also note that the above worked example further reinforces the effects of on the exponential graph.

The effects of , and on an exponential graph.

On the same set of axes, plot the following graphs (, and changes):

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs (, and changes):

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs (, and changes).

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).

For , the graph is shifted vertically upwards by units.

For , the graph is shifted vertically downwards by units.

The horizontal asymptote is shifted by units and is the line .

The effect of

The sign of determines whether the graph curves upwards or downwards.

For :

For , the graph curves downwards. It reflects the graph about the horizontal asymptote.

For , the graph curves upwards.

For :

For , the graph curves upwards.

For , the graph curves downwards. It reflects the graph about the horizontal asymptote.

The effect of and on an exponential graph when .

The effect of and on an exponential graph when .

Discovering the characteristics

The standard form of an exponential function is .

Domain and range

For , the function is defined for all real values of . Therefore the domain is .

The range of is dependent on the sign of .

For :

Therefore, for the range is .

For :

Therefore, for the range is .

Worked example 13: Domain and range of an exponential function

Find the domain and range of

Find the domain

The domain of is .

Find the range

Therefore the range is .

Intercepts

The -intercept.

For the -intercept, let :

For example, the -intercept of is given by setting :

This gives the point .

The -intercept:

For the -intercept, let .

For example, the -intercept of is given by setting :

There is no real solution. Therefore, the graph of does not have any -intercepts.

Asymptotes

Exponential functions of the form have a single horizontal asymptote, the line .

Sketching graphs of the form

In order to sketch graphs of functions of the form, , we need to determine four characteristics:

sign of

-intercept

-intercept

asymptote

The following video shows some examples of sketching exponential functions.

Video: 2FYW

Worked example 14: Sketching an exponential function

Sketch the graph of . Mark the intercept and the asymptote.

Examine the standard form of the equation

From the equation we see that , therefore the graph curves upwards. therefore the graph is shifted vertically upwards by units.

Calculate the intercepts

For the -intercept, let :

This gives the point .

For the -intercept, let :

There is no real solution, therefore there is no -intercept.

Determine the asymptote

The horizontal asymptote is the line .

Plot the points and sketch the graph

Domain:

Range:

Note that there is no axis of symmetry for exponential functions.

Worked example 15: Sketching an exponential graph

Sketch the graph of

Examine the standard form of the equation

From the equation we see that therefore the graph curves downwards. therefore the graph is shifted vertically upwards by units.

Calculate the intercepts

For the -intercept, let :

This gives the point .

For the -intercept, let :

This gives the point .

Determine the asymptote

The horizontal asymptote is the line .

Plot the points and sketch the graph

Domain:

Range:

Exercise 6.5

Given the following equation:

Calculate the -intercept. Your answer must be correct to 2 decimal places.

The -intercept is .

Now calculate the -intercept. Estimate your answer to one decimal place if necessary.

We calculate the -intercept by letting . Then we solve for :

To find the answer we try different values of :

We can see that the exponent must be between 0 and 1. Next we try values starting with and see what the value of the exponent is. Doing this we find that .

The -intercept is .

The graph here shows an exponential function with the equation . One point is given on the curve: Point A is at . Determine the values of and , correct to the nearest integer.

The asymptote lies at . Therefore is 4.

At this point we know that the equation for the graph must be .

Therefore

Below you see a graph of an exponential function with the equation . One point is given on the curve: Point A is at . Calculate the values of and , correct to the nearest integer.

The asymptote lies at . Therefore is 5.

At this point we know that the equation for the graph must be .

Therefore .

Given the following equation:

Calculate the -intercept. Your answer must be correct to 2 decimal places.

Therefore the -intercept is .

Now calculate the -intercept.

We calculate the -intercept by letting . Then start to solve for .

Therefore the -intercept is .

Given the following graph, identify a function that matches each of the following equations:

Given the functions and .

Draw the graphs on the same set of axes.

For :

is positive and greater than 1 and so the graph curves upwards. The -intercept is . There is no -intercept. The asymptote is the line .

For :

is positive and less than 1 and so the graph curves downwards. The -intercept is . There is no -intercept. The asymptote is the line .

The graph is:

Is the -axis an asymptote or an axis of symmetry to both graphs? Explain your answer.

The -axis is an asymptote to both graphs because both approach the -axis but never touch it.

Which graph can be described by the equation ? Explain your answer.

can be described by the equation because .

Solve the equation graphically and check your answer is correct by using substitution.

The graphs intersect at the point . If we substitute these values into each side of the equation we get:

LHS = RHS, therefore the answer is correct.

The curve of the exponential function in the accompanying diagram cuts the -axis at the point and passes through the point .

Determine the equation of the function .

The general form of the equation is .

We are given and .

The asymptote is at and so .

Substitute in the values of point :

Substitute in the values of point :

Therefore the equation is .

Determine the equation of the function , the reflection of in the -axis.

Determine the range of .

Range:

Determine the equation of the function , the reflection of in the -axis.

Determine the equation of the function if is a vertical stretch of by units.

Determine the equation of the function if is a vertical shift of by units.

6.6 Trigonometric functions

This section describes the graphs of trigonometric functions.

Sine function

Functions of the form

Worked example 16: Plotting a sine graph

Use your calculator to complete the following table.

Choose an appropriate scale and plot the values of on the -axis and of on the -axis. Round answers to decimal places.

Substitute values for

Plot the points and join with a smooth curve

Notice the wave shape of the graph. Each complete wave takes to complete. This is called the period. The height of the wave above and below the -axis is called the graph's amplitude. The maximum value of is and the minimum value is .

Domain:

Range:

-intercepts: , ,

-intercept:

Maximum turning point:

Minimum turning point:

Functions of the form

The effects of and on a sine graph

In the equation, , and are constants and have different effects on the graph. On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because the whole sine graph shifts up or down by units.

For , the graph is shifted vertically upwards by units.

For , the graph is shifted vertically downwards by units.

The effect of

The value of affects the amplitude of the graph; the height of the peaks and the depth of the troughs.

For , there is a vertical stretch and the amplitude increases.

For , the amplitude decreases

For , there is a reflection about the -axis.

For , there is a reflection about the -axis and the amplitude decreases.

For , there is a reflection about the -axis and the amplitude increases.

Note that amplitude is always positive.

Effect of

: vertical stretch, amplitude increases

: basic sine graph

: vertical contraction, amplitude decreases

: reflection about -axis of

: reflection about -axis of

The effect of on a sine graph.

Effect of

: vertical shift upwards by units

: basic sine graph

: vertical shift downwards by units

The effect of on a sine graph.

Discovering the characteristics

Domain and range

For , the domain is

The range of depends on the values of and .

For :

For all values of , is always between and .

Therefore for , the range of is

Similarly, for , the range of is

Period

The period of is . This means that one sine wave is completed in .

Intercepts

The -intercept of is simply the value of at

This gives the point

Important: when sketching trigonometric graphs, always start with the basic graph and then consider the effects of and .

Worked example 17: Sketching a sine graph

Sketch the graph of for .

Examine the standard form of the equation

From the equation we see that so the graph is stretched vertically. We also see that so the graph is shifted vertically upwards by units.

Substitute values for

Plot the points and join with a smooth curve

Domain:

Range:

-intercepts: none

-intercepts:

Maximum turning point:

Minimum turning point:

Cosine function

Functions of the form

Worked example 18: Plotting a cosine graph

Use your calculator to complete the following table.

Choose an appropriate scale and plot the values of on the -axis and on the -axis. Round your answers to decimal places.

Substitute values for

Plot the points and join with a smooth curve

Notice the similar wave shape of the graph. The period is also and the amplitude is . The maximum value of is and the minimum value is .

Domain:

Range:

-intercepts: ,

:

Maximum turning points: ,

Minimum turning point:

Functions of the form

The effects of and on a cosine graph

In the equation, , and are constants and have different effects on the graph.

On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because the whole cosine graph shifts up or down by units.

For , the graph is shifted vertically upwards by units.

For , the graph is shifted vertically downwards by units.

The effect of

The value of affects the amplitude of the graph; the height of the peaks and the depth of the troughs.

For , there is a vertical stretch and the amplitude increases.

For , the amplitude decreases.

For , there is a reflection about the -axis.

For , there is a reflection about the -axis and the amplitude decreases.

For , there is a reflection about the -axis and the amplitude increases.

Note that amplitude is always positive.

Effect of

: vertical stretch, amplitude increases

: basic cosine graph

: amplitude decreases

: reflection about -axis, amplitude decreases

: reflection about -axis, amplitude increases

The effect of on a cosine graph.

Effect of

: vertical shift upwards by units

: basic cosine graph

: vertical shift downwards by units

The effect of on a cosine graph.

Discovering the characteristics

Domain and range

For , the domain is

It is easy to see that the range of will be the same as the range of . This is because the maximum and minimum values of will be the same as the maximum and minimum values of .

For the range of is

For the range of is

Period

The period of is . This means that one cosine wave is completed in .

Intercepts

The -intercept of is calculated in the same way as for sine.

This gives the point .

Worked example 19: Sketching a cosine graph

Sketch the graph of for .

Examine the standard form of the equation

From the equation we see that so the graph is stretched vertically. We also see that so the graph is shifted vertically upwards by units.

Substitute values for

Plot the points and join with a smooth curve

Domain:

Range:

-intercepts: none

-intercept:

Maximum turning points: ,

Minimum turning point:

Comparison of graphs of and

Notice that the two graphs look very similar. Both waves move up and down along the -axis. The distances between the peaks for each graph is the same. The height of the peaks and the depths of the troughs are also the same.

If you shift the whole cosine graph to the right by it will overlap perfectly with the sine graph. If you shift the sine graph to the left it would overlap perfectly with the cosine graph. This means that:

Tangent function

Functions of the form

Worked example 20: Plotting a tangent graph

Use your calculator to complete the following table.

Choose an appropriate scale and plot the values with on the -axis and on the -axis. Round your answers to decimal places.

Substitute values for

undef

undef

Plot the points and join with a smooth curve

There is an easy way to visualise the tangent graph. Consider our definitions of and for right-angled triangles:

So for any value of :

So we know that for values of for which , we must also have . Also if the value of is undefined as we cannot divide by . The dashed vertical lines are at the values of where is not defined and are called the asymptotes.

Asymptotes: the lines and

Period:

Domain:

Range:

-intercepts: , ,

-intercept:

Functions of the form

The effects of and on a tangent graph

On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

On the same set of axes, plot the following graphs for :

Use your results to deduce the effect of .

The effect of

The effect of is called a vertical shift because the whole tangent graph shifts up or down by units.

For , the graph is shifted vertically upwards by units.

For , the graph is shifted vertically downwards by units.

The effect of

The value of affects the steepness of each of the branches of the graph. The greater the value of , the quicker the branches of the graph approach the asymptotes.

The effect of and on a tangent graph.

Discovering the characteristics

Domain and range

From the graph we see that is undefined at and .

Therefore the domain is .

The range is .

Period

The period of is . This means that one tangent cycle is completed in .

Intercepts

The -intercept of is simply the value of at .

This gives the point .

Asymptotes

The graph has asymptotes at and .

Worked example 21: Sketching a tangent graph

Sketch the graph of for .

Examine the standard form of the equation

We see that so the branches of the curve will be steeper. We also see that so the graph is shifted vertically upwards by unit.

Substitute values for

–

–

Plot the points and join with a smooth curve

Domain:

Range:

Exercise 6.6

Shown the following graph of the following form: where Point A is at , and Point B is at , find the values of and .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted sine graph point would be at . For this graph we see that this point has been shifted up by or spaces. Therefore .

To find we note that the -value at the middle (point ) is , while the -value at the top (point ) is 3. We can find the amplitude by working out the distance from the top of the graph to the middle of the graph: . Therefore .

The complete equation for the graph shown in this question is .

Therefore

Shown the following graph of the following form: where Point A is at , and Point B is at , determine the values of and .

To find we note that the -value at the bottom (point ) is , while the -value at the top (point ) is 2. We can find the amplitude by working out the distance from the top of the graph to the bottom of the graph and then dividing this by 2 since this distance is twice the amplitude: . Therefore .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted sine graph with the same value (i.e. ) point would be at . For this graph we see that this point has been shifted down by spaces. Therefore .

The complete equation for the graph shown in this question is .

Therefore

The graph below shows a trigonometric equation of the following form: . Two points are shown on the graph: Point A at , and Point B: . Calculate the values of (the amplitude of the graph) and (the vertical shift of the graph).

To find we note that the -value at the bottom (point ) is , while the -value at the top (point ) is . We can find the amplitude by working out the distance from the top of the graph to the bottom of the graph and then dividing this by 2 since this distance is twice the amplitude: . Therefore .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted cosine graph with the same value (i.e. ) point would be at . For this graph we see that this point has been shifted down by space. Therefore .

The complete equation for the graph shown in this question is .

Therefore .

The graph below shows a trigonometric equation of the following form: . Two points are shown on the graph: Point A at , and Point B: . Calculate the values of (the amplitude of the graph) and (the vertical shift of the graph).

To find we note that the -value at the bottom (point ) is , while the -value at the middle (point ) is . We can find the amplitude by working out the distance from the top of the graph to the middle of the graph: . Therefore .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted cosine graph with the same value (i.e. ) point would be at . For this graph we see that this point has not been shifted. Therefore .

The complete equation for the graph shown in this question is .

Therefore .

On the graph below you see a tangent curve of the following form: . Two points are labelled on the curve: Point A is at , and Point B is at .

Calculate, or otherwise determine, the values of and .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted tangent graph point would be at . For this graph we see that this point been shifted upwards by a . Therefore .

To find we can substitute point into the equation for the tangent graph:

The complete equation is: .

Therefore .

The graph below shows a tangent curve with an equation of the form . Two points are labelled on the curve: Point A is at , and Point B is at .

Find and .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted tangent graph point would be at . For this graph we see that the graph has not been shifted. Therefore .

To find we can substitute point into the equation for the tangent graph:

The complete equation is: .

Therefore .

Given the following graph, identify a function that matches each of the following equations:

The graph below shows functions and

What is the equation for ?

With the assistance of the table below sketch the three functions on the same set of axes.

We are given a table with values and so we plot each of these points and join them with a smooth curve.

With the assistance of the table below sketch the three functions on the same set of axes.

We are given a table with values and so we plot each of these points and join them with a smooth curve.

State the coordinates at and the range of the function.

To find the coordinates of we read the value off the graph. To find the range we note that this is a cosine graph and so the maximum value occurs at (and at ). The minimum value occurs at . So we read off the value of at and at .

Therefore .

To find the coordinates of we read the value off the graph. To find the range we note that this is a sine graph and so the maximum value occurs at . The minimum value occurs at . So we read off the value of at and at .

Therefore

To find the coordinates of we read the value off the graph. To find the range we note that this is a sine graph and so the maximum value occurs at . The minimum value occurs at . So we read off the value of at and at .

Therefore

To find the coordinates of we read the value off the graph. To find the range we note that this is a cosine graph and so the maximum value occurs at (and at ). The minimum value occurs at . So we read off the value of at and at .

Therefore

State the coordinates at and the domain and range of the function in the interval shown.

Using your knowledge of the effects of and , sketch each of the following graphs, without using a table of values, for

In this case and so the basic sine graph is not shifted up or downwards. We also note that and so the graph is stretched by 2 units. The maximum value will be 2 and the minimum value will be .

In this case and so the basic cosine graph is not shifted up or downwards. We also note that and so the graph is stretched by units. The maximum value will be 4 and the minimum value will be .

In this case and so the basic cosine graph is shifted upwards by 1 unit. We also note that and so the graph is stretched by units. The maximum value will be 3 and the minimum value will be .

In this case and so the basic sine graph is shifted downwards by 3 units. We also note that and so the graph is not stretched. The maximum value will be and the minimum value will be .

In this case and so the basic tangent graph is shifted downwards by 2 units. We also note that and so the graph is not stretched. When , . Similarly when , and when , .

In this case and so the basic cosine graph is shifted downwards by 1 units. We also note that and so the graph is stretched by 2 units. The maximum value will be and the minimum value will be .

Give the equations for each of the following graphs:

The general form of a cosine graph is . We note that in this case the graph is not shifted. We also note the graph is stretched by units.

Therefore .

The general form of a sine graph is . We note that in this case the graph is shifted upwards by 1 unit. We also note the graph is not stretched.

Therefore .

For which values of is the function increasing, in the interval shown?

For which values of is the function negative, in the interval shown?

For which values of is the function positive, in the interval shown?

Given the following graph.

State the coordinates at and .

We can read the values off the graph:

How many times in this interval does intersect .

What is the amplitude of .

Evaluate: .

Read off the value of and from the graph. Then subtract from .

Given the following graph.

State the coordinates at and .

We read the values off the graph:

How many times in this interval does intersect .

What is the amplitude of .

Evaluate: .

Read off the value of and from the graph. Then subtract from .

Given the following graph:

State the coordinates at and .

We read the values off the graph:

How many times in this interval does intersect .

What is the amplitude of .

Evaluate: .

Read off the value of and from the graph. Then subtract from .

6.7 Interpretation of graphs

Worked example 22: Determining the equation of a parabola

Use the sketch below to determine the values of and for the parabola of the form .

Examine the sketch

From the sketch we see that the shape of the graph is a “frown”, therefore . We also see that the graph has been shifted vertically upwards, therefore .

Determine using the -intercept

The -intercept is the point .

Use the other given point to determine

Substitute point into the equation:

Write the final answer

and , so the equation of the parabola is .

Worked example 23: Determining the equation of a hyperbola

Use the sketch below to determine the values of and for the hyperbola of the form .

Examine the sketch

The two curves of the hyperbola lie in the second and fourth quadrant, therefore . We also see that the graph has been shifted vertically upwards, therefore .

Substitute the given points into the equation and solve

Substitute the point :

Substitute the point :

Solve the equations simultaneously using substitution

Write the final answer

and , therefore the equation of the hyperbola is .

Worked example 24: Interpreting graphs

The graphs of and are given. Calculate the following:

coordinates of , , ,

coordinates of

distance

Calculate the intercepts

For the parabola, to calculate the -intercept, let :

This gives the point .

To calculate the -intercept, let :

This gives the points and .

For the straight line, to calculate the -intercept, let :

This gives the point .

Calculate the point of intersection

At the two graphs intersect so we can equate the two expressions:

At , , therefore . This gives the point .

Calculate distance

Distance is units.

Write the final answer

coordinates of , , ,

coordinates of

distance units

Worked example 25: Interpreting trigonometric graphs

Use the sketch to determine the equation of the trigonometric function of the form .

Examine the sketch

From the sketch we see that the graph is a sine graph that has been shifted vertically upwards. The general form of the equation is .

Substitute the given points into equation and solve

At , and :

At , and :

Solve the equations simultaneously using substitution

Write the final answer

Exercise 6.7

Plot the following functions on the same set of axes and clearly label all the points at which the functions intersect.

and

The -intercept for each graph is:

This is also the only point of intersection.

For both graphs there is no -intercept.

and

is a basic straight line graph. For there is no -intercept and no -intercept. We note that this is a hyperbolic graph that has been stretched by 2 units.

The points of intersection are:

The graphs intersect at and .

and

is a horizontal line through . For the -intercept is and there are no -intercepts. From the value of we see that this is a basic parabola that has been shifted upwards by 3 units.

The points of intersection are:

The graphs intersect at and .

and

is a parabola that has been reflected about the -axis. For there is no -intercept and there is no -intercepts. From the value of we see that this is a basic hyperbola that has been stretched by 8 units.

The points of intersection are:

The graphs intersect at .

Determine the equations for the graphs given below.

For the straight line graph we have the and -intercepts. The -intercept gives . Now we can calculate the gradient of the straight line graph:

Therefore the equation of the straight line graph is .

For the parabola we also have the and -intercepts. The -intercept gives . Now we can calculate :

Therefore the equation of the parabola is .

The equations for the two graphs are and

For the straight line graph we notice that it passes through and so .

We have two points on the straight line graph and so we can calculate the gradient, :

The equation of the straight line graph is .

For the hyperbola we note that the graph is not shifted either upwards or downwards. Therefore . Now we can calculate :

Therefore the equation of the hyperbola is .

The equations for the two graphs are and .

Choose the correct answer:

The range of is:

(iii)

The range of is:

(iv)

The y-intercept of is:

(iii)

Which of the following passes through ?

(i)

6.8 Chapter summary

Presentation: 2G2K

Characteristics of functions:

The given -value is known as the independent variable, because its value can be chosen freely. The calculated -value is known as the dependent variable, because its value depends on the -value.

The domain of a function is the set of all -values for which there exists at most one -value according to that function. The range is the set of all -values, which can be obtained using at least one -value.

An asymptote is a straight line, which the graph of a function will approach, but never touch.

Special functions and their properties:

Linear functions of the form .

Parabolic functions of the form .

Hyperbolic functions of the form .

Exponential functions of the form .

Trigonometric functions of the form

Exercise 6.8

Complete the following tables and identify the function.

Plot the following points on a graph.

Create a table of values from the function given and then plot the function. Your table must have at least 5 ordered pairs.

Determine the -intercept and the -intercepts of the function.

-intercept and -intercept

-intercept and -intercept

The graph below shows an equation, which has the form . Calculate or otherwise find the values of (the gradient of the line) and (the -intercept of the line).

Point is the -intercept. Point has co-ordinates and so .

Therefore , and .

Look at the graphs below. Each graph is labelled with a letter. In the questions that follow, match any given equation with the label of a corresponding graph.

State whether the following is true or not

The -intercept of is .

True

The gradient of is .

False

The gradient of is .

False

Write the following in standard form:

Sketch the graphs of the following:

The -intercept is and the -intercept is .

Write the equation in standard form: .

The -intercept is and the -intercept is .

We note the following pairs of values: , , and . Now we can draw the graph.

Write the equation in standard form: .

The -intercept is and the -intercept is .

The function for how much water a tap dispenses is given by: , where are in seconds and respectively.

Use this information to answer the following:

Evaluate .

Evaluate .

How long will it take to fill a bottle of water?

How much water can the tap dispense in ?

The graph below shows the distance travelled by a car over time, where is distance in and is time in minutes.

Use this information to answer the following:

What distance did the car travel in an hour?

50km

What is the domain of the function?

The domain is min.

What is the range of the function? What does it represent?

The range is km, it represents the total distance travelled.

On the graph here you see a function of the form: . Two points on the parabola are shown: Point A, the turning point of the parabola, at , and Point B is at . Calculate the values of and .

The value of is 6.

Given the following equation:

Calculate the -coordinate of the -intercept.

The -coordinate of the -intercept is .

Now calculate the -intercepts. Your answer must be correct to 2 decimal places.

The -intercepts are and .

Given the following graph, identify a function that matches each of the given equations:

Given the following graph, identify a function that matches each of the given equations:

Sketch the following functions:

The -intercept is . There are no -intercepts.

is positive and so the graph is a smile with a minimum turning point at .

The -intercept is . There are no -intercepts.

is positive and so the graph is a smile with a minimum turning point at .

The -intercept is . The -intercepts are at and .

is positive and so the graph is a smile with a minimum turning point at .

Sebastian and Lucas dive into a pool from different heights. Their midair paths can be described by the following quadratic equations: for Sebastian and for Lucas.

From what height did Sebastian dive?

Maximum value of is

From what height did Lucas dive?

Maximum value of is

How far from the pool wall did Lucas land?

Lucas landed from the pool wall.

How much closer to the pool wall did Sebastian land compared to Lucas?

Sebastian landed from the pool wall.

Therefore Sebastian landed closer to the wall than Lucas.

The following graph shows a hyperbolic equation of the form . Point A is shown at . Calculate the values of and .

Therefore and .

The equation is .

Given the following equation:

Determine the location of the -intercept.

There is no -intercept.

Determine the location of the -intercept.

The -intercept is at .

Given the following graph, identify a function that matches each of the given equations:

Sketch the following functions and identify the asymptotes:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept. The -intercept is at .

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept. The -intercept is at .

Sketch the functions given and describe the transformation used to obtain the second function. Show all asymptotes.

and

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept. The -intercept is at .

Translation by 2 in the positive -direction.

and

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

Reduction by 4

and

First simplify the second equation:

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept. The -intercept is at .

Translation by 3 units in the positive -direction.

and

:

The asymptote is .

is positive and so the graph lies in the first and third quadrants.

There is no -intercept and no -intercept.

:

The asymptote is .

is negative and so the graph lies in the second and fourth quadrants.

There is no -intercept and no -intercept.

Reflection on -axis

Given the following equation:

Calculate the -intercept. Your answer must be correct to 2 decimal places.

The -intercept is .

Now calculate the -intercept. Estimate your answer to one decimal place if necessary.

We calculate the -intercept by letting . Then start to solve for .

We can see that the exponent must be between 1 and 2. By trial and error we get . Therefore the -intercept is .

Sketch the following functions and identify the asymptotes:

The -intercept is . There is no -intercept. The asymptote is at .

therefore the graph curves upwards.

The -intercept is . There is no -intercept. The asymptote is at .

therefore the graph curves downwards.

The -intercept is . The -intercept is . The asymptote is at .

therefore the graph curves downwards.

The form of the curve graphed below is . One point is given on the curve: Point A is at . Find the values of and , correct to the nearest integer.

The asymptote lies at . Therefore is .

At this point we know that the equation for the graph must be .

Given the following graph, identify a function that matches each of the given equations

Use the functions to find the value of the following:

Determine whether the following statements are true or false. If the statement is false, give reasons why.

The given or chosen -value is known as the independent variable.

False, the given or chosen -value is the dependent variable because it's value depends on the independent variable .

A graph is said to be continuous if there are breaks in the graph.

False, a graph is said to be continuous if there are no breaks in it.

Functions of the form are straight lines.

True

Functions of the form are exponential functions.

False, functions of the form are hyperbolic functions.

An asymptote is a straight line which a graph will intersect at least once.

False, an asymptote is a straight line that a graph will never intersect.

Given a function of the form , to find the -intercept let and solve for .

True

Given the functions and .

Draw and on the same set of axes.

For the -intercept is and the -intercept is .

For the -intercept is and the -intercepts are and .

Calculate the points of intersection of and .

The -values of the points of intersection can be found by setting :

The -values can be obtained by substituting the -values into either equation:

Therefore the points of intersection are and .

Use your graphs and the points of intersection to solve for when:

Therefore, for , .

Therefore, for , .

This is found by looking at where the graph of lies underneath the graph of .

For , .

Give the equation of the reflection of in the -axis.

After a ball is dropped, the rebound height of each bounce decreases. The equation shows the relationship between the number of bounces and the height of the bounce for a certain ball. What is the approximate height of the fifth bounce of this ball to the nearest tenth of a unit?

For the fifth bounce . Now we can solve for :

Therefore the approximate height of the fifth bounce is units

Mark had coins in and pieces. He had more coins than coins. He wrote a system of equations to represent this situation, letting represent the number of coins and represent the number of coins. Then he solved the system by graphing.

Write down the system of equations.

Let coins and coins. Then the system of equations is:

;

Draw their graphs on the same set of axes.

For the -intercept is and the -intercept is .

For the -intercept is and the -intercept is .

Use your sketch to determine how many and pieces Mark had.

From the sketch we see that the graphs intersect at . Checking algebraically we get:

Substitute the value of into the second equation:

Substitute the value of back into the first equation:

Mark has coins and coins.

Shown the following graph of the following form: where Point A is at , and Point B is at , determine the values of and .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted sine graph point would be at . For this graph we see that this point has been shifted up by 3 spaces. Therefore .

To find we note that the -value at the middle (point ) is 3, while the -value at the top (point ) is . We can find the amplitude by working out the distance from the top of the graph to the middle of the graph: . Therefore .

The complete equation for the graph shown in this question is .

Therefore .

The graph below shows a trigonometric equation of the following form: . Two points are shown on the graph: Point A at , and Point B: . Calculate the values of (the amplitude of the graph) and (the vertical shift of the graph).

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted cosine graph point would be at . For this graph we see that this point has not been shifted. Therefore .

To find we note that the -value at the middle (point ) is 0, while the -value at the bottom (point ) is . We can find the amplitude by working out the distance from the middle of the graph to the bottom of the graph: . Therefore .

The complete equation for the graph shown in this question is .

Therefore .

On the graph below you see a tangent curve of the following form: . Two points are labelled on the curve: Point A is at , and Point B is at .

Calculate, or otherwise determine, the values of and .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance point is at . For an unshifted tangent graph point would be at . For this graph we see that this point been shifted downwards by . Therefore .

To find we can substitute point into the equation for the tangent graph:

The complete equation is: .

Therefore .

Given the following graph, identify a function that matches each of the given equations:

The graph below shows functions and .

What is the equation for ?

With the assistance of the table below sketch the three functions on the same set of axes.

With the assistance of the table below sketch the three functions on the same set of axes.

Sketch graphs of the following trigonometric functions for . Show intercepts and asymptotes.

The -intercept is . The -intercepts are and . There are no asymptotes.

The graph is not shifted up or down since . The graph is stretched by 4 and reflected in the -axis.

The -intercept is . There are no -intercepts. There are no asymptotes.

The graph is shifted down by since . The graph is not stretched since .

The -intercept is . The -intercepts are and . There are no asymptotes.

The graph is shifted up by 1 since . The graph is stretched by and reflected in the -axis since .

The -intercept is . The -intercepts are and . The asymptotes are and .

The graph is shifted up by 2 since . The graph is not stretched since .

The -intercept is . The -intercepts are and . There are no asymptotes.

The graph is not shifted upwards or downwards since . The graph is stretched by since .

State the coordinates at and the range of the function.

State the coordinates at and the domain and range of the function in the interval shown.

For which values of is the function decreasing, in the interval shown?

For which values of is the function increasing, in the interval shown?

For which values of is the function positive, in the interval shown?

Given the general equations , , , , , and , determine the specific equations for each of the following graphs.

This is a straight line graph and so the general equation is . The -intercept is at and so .

To find we substitute in the given point into the equation and solve for :

Therefore the equation is .

This is a parabola and so we use . The -intercept is at and so .

We substitute the point into the equation and solve for :

Therefore the equation is .

This is a hyperbola and so we use . There is no -intercept and so the graph has not been shifted upwards or downwards. Therefore .

We substitute the point into the equation and solve for :

Therefore the equation is .

This is a straight line graph and so the general equation is . The -intercept is at and so .

To find we substitute the point into the equation and solve for :

Therefore the equation is .

This is a sine graph and so the general equation is .

To find we note that the -value at the bottom is , while the -value at the top is . We can find the amplitude by working out the distance from the bottom of the graph to the top of the graph and dividing this value by 2: . Therefore .

To find we note that shifts the graph up or down. To determine we can look at any point on the graph. For instance we can see that when , . For an unshifted sine graph with the same value (i.e. ) this point would be at . For this graph we see that this point has been shifted upwards by 1 unit. Therefore .

The complete equation for the graph shown in this question is .

This is an exponential graph and so we use . We see that the asymptote is at and so . To find we substitute the point into the equation:

To find we substitute the point into the equation:

Therefore the equation is .

This is a tangent graph and so we use . To find we note that the graph has been shifted down by 2 units (the point is given). Therefore .

To find we substitute into the equation:

Therefore the equation is .

State the coordinates at and .

How many times in this interval does intersect .

0

What is the amplitude of .

1

Evaluate: .

State the coordinates at and .

How many times in this interval does intersect .

2

What is the amplitude of .

3

Evaluate: .

and are sketched below. Answer the questions that follow.

Calculate the coordinates of and .

is the -intercept of and so . Therefore the coordinates of are .

is the -intercept of and so . Therefore the coordinates of are .

Therefore and .

Calculate the length of .

and both lie on the -axis and so they both lie on a straight line.

Therefore .

Calculate the length of if .

At , , therefore .

At , , therefore .

Therefore length .

Give the equation of reflected about the -axis.

Give the range of both graphs.

Range :

Range :

Plot the following functions on the same set of axes and clearly label all points of intersection.

For :

The -intercept is at . The -intercepts are at and .

For :

The -intercept is at . The -intercept is at .

There are no points of intersection.

For :

The -intercept is at . The -intercepts are at and .

For :

The -intercept is at . The -intercept is at .

To find the point of intersection we equate the two functions:

Therefore the graphs intersect at and .

and are sketched below. The points and are given. Answer the questions that follow.

Determine the value of .

Point is the -intercept of and so .

Calculate the length of .

is at and so is at . To find we substitute point into :

Therefore .

Give the equation of reflected about the -axis.

Give the equation of shifted vertically upwards by unit.

Give the equation of the asymptote of .

Give the ranges of and .

Range : , Range :

Given and . Answer the questions that follow.

Sketch both graphs on the same set of axes.

For the -intercept is at . The -intercepts are at and .

For the -intercept is at . The -intercepts are at and .

Describe the relationship between and .

is therefore the reflection of about the -axis.

Give the equation of reflected about the line .

Give the domain and range of .

Domain : . Range : .

Sketch the graphs of and on the same set of axes. Use your sketch to determine:

The domain and range of .

The amplitude and period of .

Domain: . Range:

Amplitude: . Period:

.

The graphs of and are shown in the following diagram.

Calculate:

The coordinates of points and .

and are the points of intersection of the two functions. Therefore:

Since the equation of the straight line is these are also the -values of the points of intersection.

Therefore and

The length of .

has the same value as and has the same value as .

Therefore and .

.

The length of .

Using Pythagoras:

The length of , given .

and have the same value as point . lies on and so . lies on and so .

Therefore length units.

Given the diagram with and .

Calculate the coordinates of , and .

and are the -intercepts of . is the -intercept of .

Therefore point is at .

Points and are at and respectively.

Therefore , ,

Describe in words what happens at point .

The parabola and the hyperbola intersect at point which lies in the fourth quadrant.

Calculate the coordinates of .

Determine the equation of the straight line that would pass through points and .

Determine gradient and :

is the -intercept and so .

Therefore .

The diagram shows the graphs of and

Give the domain of .

Domain:

What is the amplitude of ?

Amplitude:

Determine for which values of :

is increasing

Determine the equations for the graphs given below.

For the straight line:

For the parabola:

Therefore the equations are: en .

For the straight line:

For the exponential graph:

Therefore the equations are and .

Choose the correct answer:

Which of the following does not have a gradient of 3?

(iv)

The asymptote of is:

(ii)

Sketch the following

The asymptote is at . The -intercept is at . There is no -intercept.

First rewrite the equation in standard form:

There is no -intercept. The -intercept is at . The asymptote is at .

First write the equation in standard form:

Chapter 7: Euclidean geometry

Content covered in this chapter includes revision of lines, angles and triangles. The mid-point theorem is introduced. Kites, parallelograms, rectangle, rhombus, square and trapezium are investigated.

Solving problems and proving riders is only covered later in the year. The focus of this chapter is on introducing the special quadrilaterals and revising content from earlier grades.

Revision of triangles should focus on similar and congruent triangles.

Sketches are valuable and important tools. Encourage learners to draw accurate diagrams to solve problems.

It is important to stress to learners that proportion gives no indication of actual length. It only indicates the ratio between lengths.

Notation - emphasise to learners the importance of the correct ordering of letters, as this indicates which angles are equal and which sides are in the same proportion.

Geometry (from the Greek “geo” = earth and “metria” = measure) arose as the field of knowledge dealing with spatial relationships. Analytical geometry deals with space and shape using algebra and a coordinate system. Euclidean geometry deals with space and shape using a system of logical deductions.

Euclidean geometry was first used in surveying and is still used extensively for surveying today. Euclidean geometry is also used in architecture to design new buildings. Other uses of Euclidean geometry are in art and to determine the best packing arrangement for various types of objects.

Figure 7.1: A small piece of the original version of Euclid's elements. Euclid is considered to be the father of modern geometry. Euclid's elements was used for many years as the standard text for geometry.

This video highlights some of the basic concepts used in geometry.

Video: 2G5V

In Euclidean geometry we use two fundamental types of measurement: angles and distances.

GeoGebra is a useful tool to use for sketching out the worked examples and activities.

7.1 Introduction

Angles

An angle is formed when two straight lines meet at a point, also known as a vertex. Angles are labelled with a caret on a letter, for example, . Angles can also be labelled according to the line segments that make up the angle, for example or . The symbol is a short method of writing angle in geometry and is often used in phrases such as “sum of s in ”. Angles are measured in degrees which is denoted by , a small circle raised above the text, similar to an exponent.

Properties and notation

In the diagram below two straight lines intersect at a point, forming the four angles , , and .

The following table summarises the different types of angles, with examples from the figure above.

Term

Property

Examples

Acute angle

;

Right angle

Angle

Obtuse angle

;

Straight angle

Angle

;

Reflex angle

Adjacent angles

Angles that share a vertex and a common side.

and ; and

Vertically opposite angles

Angles opposite each other when two lines intersect. They share a vertex and are equal.

;

Supplementary angles

Two angles that add up to

;

Complementary angles

Two angles that add up to

Revolution

The sum of all angles around a point.

Note that adjacent angles on a straight line are supplementary.

The following video provides a summary of the terms used to refer to angles.

Video: 2G5W

Parallel lines and transversal lines

Two lines intersect if they cross each other at a point. For example, at a traffic intersection two or more streets intersect; the middle of the intersection is the common point between the streets.

Parallel lines are always the same distance apart and they are denoted by arrow symbols as shown below.

In writing we use two vertical lines to indicate that two lines are parallel:

A transversal line intersects two or more parallel lines. In the diagram below, and is a transversal line.

The properties of the angles formed by these intersecting lines are summarised in the following table:

Name of angle

Definition

Examples

Notes

Interior angles

Angles that lie in between the parallel lines.

, , and are interior angles.

Interior means inside.

Exterior angles

Angles that lie outside the parallel lines.

, , and are exterior angles.

Exterior means outside.

Corresponding angles

Angles on the same side of the lines and the same side of the transversal. If the lines are parallel, the corresponding angles will be equal.

and , and , and , and are pairs of corresponding angles. , , and .

Co-interior angles

Angles that lie in between the lines and on the same side of the transversal. If the lines are parallel, the angles are supplementary.

and , and are pairs of co-interior angles. , .

Alternate interior angles

Equal interior angles that lie inside the lines and on opposite sides of the transversal. If the lines are parallel, the interior angles will be equal.

and , and are pairs of alternate interior angles. ,

This video provides a short summary of some of the angles formed by intersecting lines.

Video: 2G5X

If two lines are intersected by a transversal such that:

corresponding angles are equal; or

alternate interior angles are equal; or

co-interior angles are supplementary

then the two lines are parallel.

When we refer to lines we can either write to mean the line through points and or to mean the line segment from point to point .

Worked example 1: Finding angles

Find all the unknown angles. Is ? Explain your answer.

Use the properties of parallel lines to find all equal angles on the diagram

Redraw the diagram and mark all the equal angles.

Determine the unknown angles

Determine whether

If then will be equal to corresponding angle , but and . Therefore is not parallel to .

Exercise 7.1

Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labelled with letters in the diagram:

You can redraw the diagram and fill in the angles as you find them.

Find all the unknown angles in the figure:

Find the value of in the figure:

(corresp s; ).

(corresp s; ).

Find each of the unknown angles marked in the figure below. Find a reason that leads to the answer in a single step.

and are alternate interior angles on transversal . Therefore, they must be equal in size since .

Therefore .

We have just found that . (s on a str line)

and are corresponding angles ().

Therefore: .

(s on a str line).

(vert opp s )

Therefore: .

Based on the results for the angles above, is ?

To prove we need to show that one of the following is true:

(corresp s)

(alt s)

(co-int s)

However , therefore is not parallel to .

Find each of the unknown angles marked in the figure below. Find a reason that leads to the answer in a single step.

and are alternate interior angles on transversal . Since they must be equal in size.

Therefore .

We have just found that . (s on a str line)

and are corresponding angles ().

Therefore: .

(s on a str line).

(vert opp s ).

Therefore: .

Based on the results for the angles above, is ?

To prove we need to show that one of the following is true:

(corresp s)

(alt s)

(co-int s)

(corresp s), therefore . We also note that and .

Determine whether the pairs of lines in the following figures are parallel:

If then (co-int s). But . Therefore there are no parallel lines, is not parallel to . Note that we do not consider as this is a transversal.

(s on a str line). If then would be equal to , is not parallel to . Note that is a transversal.

Let be point of intersection of lines and and be the point of intersection of lines and .

These are corresponding angles .

If is parallel to and is parallel to , explain why must be parallel to .

If and then we know that .

Similarly if and then we know that .

7.2 Triangles

Classification of triangles

A triangle is a three-sided polygon. Triangles can be classified according to sides: equilateral, isosceles and scalene. Triangles can also be classified according to angles: acute-angled, obtuse-angled and right-angled.

We use the notation to refer to a triangle with vertices labelled , and .

Name

Diagram

Properties

Scalene

All sides and angles are different.

Isosceles

Two sides are equal in length. The angles opposite the equal sides are also equal.

Equilateral

All three sides are equal in length and all three angles are equal.

Acute

Each of the three interior angles is less than .

Obtuse

One interior angle is greater than .

Right-angled

One interior angle is .

Different combinations of these properties are also possible. For example, an obtuse isosceles triangle and a right-angled isosceles triangle are shown below:

This video shows the different ways to classify triangles.

Video: 2G69

Interior angles of a triangle

On a piece of paper draw a triangle of any size and shape.

Cut it out and label the angles , and on both sides of the paper.

Draw dotted lines as shown and cut along these lines to get three pieces of paper.

Place them along your ruler as shown in the figure below.

What can we conclude?

Hint: What is the sum of angles on a straight line?

Exterior angles of a triangle

On a piece of paper draw a triangle of any size and shape. On another piece of paper, make a copy of the triangle.

Cut both out and label the angles of both triangles , and on both sides of the paper.

Draw dotted lines on one triangle as shown and cut along the lines.

Place the second triangle and the cut out pieces as shown in the figure below.

What can we can conclude?

We can use the fact that the angles in a triangle add up to to work out the sum of the exterior angles in a pentagon. This video shows you how.

Video: 2G6B

Congruency

Two triangles are congruent if one fits exactly over the other. This means that the triangles have equal corresponding angles and sides. To determine whether two triangles are congruent, it is not necessary to check every side and every angle. We indicate congruency using .

The following table describes the requirements for congruency:

Rule

Description

Diagram

RHS or HS

(, hypotenuse, side)

If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two triangles are congruent.

SSS

(side, side, side)

If three sides of a triangle are equal in length to the corresponding sides of another triangle, then the two triangles are congruent.

SAS or SS

(side, angle, side)

If two sides and the included angle of a triangle are equal to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.

AAS or S

(angle, angle, side)

If one side and two angles of a triangle are equal to the corresponding one side and two angles of another triangle, then the two triangles are congruent.

The order of letters when labelling congruent triangles is very important.

This notation indicates the following properties of the two triangles: , , , , and .

You might see used to show that two triangles are congruent. This is the internationally recognised symbol for congruency.

This video shows some practice examples of finding congruent triangles.

Video: 2G6C

Similarity

Two triangles are similar if one triangle is a scaled version of the other. This means that their corresponding angles are equal in measure and the ratio of their corresponding sides are in proportion. The two triangles have the same shape, but different scales. Congruent triangles are similar triangles, but not all similar triangles are congruent. We use to indicate that two triangles are similar.

The following table describes the requirements for similarity:

Rule

Description

Diagram

AAA

(angle, angle, angle)

If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.

, ,

SSS

(side, side, side)

If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

The order of letters for similar triangles is very important. Always label similar triangles in corresponding order. For example,

You might see used to show that two triangles are similar. This is the internationally recognised symbol for similarity.

The following video explains similar triangles.

Video: 2G6D

The theorem of Pythagoras

If is right-angled with , then .

Converse: If , then is right-angled with .

The following video explains the theorem of Pythagoras and shows some examples of working with the theorem of Pythagoras.

Video: 2G6F

Worked example 2: Triangles

Determine if the two triangles are congruent. Use the result to find , and .

Examine the information given for both triangles

Determine whether

In :

In and :

Determine the unknown angles and sides

In :

In :

Exercise 7.2

Calculate the unknown variables in each of the following figures.

The triangle is isosceles therefore (s opp equal sides).

is an exterior angle, therefore (ext of ).

First find . (s on a str line). Therefore .

is an exterior angle, therefore (ext of ).

Therefore and .

(vert opp s).

.

Therefore and are similar because they have the same angles.

Similar triangles have proportional sides:

From the theorem of Pythagoras we have:

We note that:

Now we can use the fact that the sides are in proportion to find and :

Therefore and .

From the theorem of Pythagoras:

Therefore and .

Given the following diagrams:

Diagram A

Diagram B

Which diagram correctly gives a pair of similar triangles?

Diagram A shows a pair of triangles with all pairs of corresponding angles equal (the same three angle markers are shown in both triangles). Diagram B shows a pair of triangles with different angles in each triangle. All six angles are different and there are no pairs of corresponding angles that are equal.

Therefore diagram A gives a pair of triangles that are similar.

Given the following diagrams:

Diagram A

Diagram B

Which diagram correctly gives a pair of similar triangles?

Diagram A shows a pair of triangles with different angles in each triangle. All six angles are different and there are no pairs of corresponding angles that are equal. Diagram B shows a pair of triangles with all pairs of corresponding angles equal (the same two angle markers are shown in both triangles and the third angle in each triangle must be equal).

Therefore diagram B gives a pair of triangles that are similar.

Have a look at the following triangles, which are drawn to scale:

Are the two triangles congruent? If so state the reason and use the correct notation to state that they are congruent.

We are not told if and or and therefore we cannot say that the sides are the same length. Also we are not given any information about the angles of the two triangles. Therefore we cannot say if the two triangles are congruent.

Have a look at the following triangles, which are drawn to scale:

Are the two triangles congruent? If so state the reason and use the correct notation to state that they are congruent.

Note that the two pairs of sides are equal, as indicated by the and . In addition, the angle between those two sides are marked as equal (this is the included angle).

Therefore, these two triangles are congruent. , reason: SAS.

State whether the following pairs of triangles are congruent or not. Give reasons for your answers. If there is not enough information to make a decision, explain why.

We have two equal sides ( and is common to both triangles) and one equal angle () but the sides do not include the known angle. The triangles therefore do not have a SAS and are therefore not congruent. (Note: is not necessarily equal to because it is not given that ).

There is not enough information given. We need at least three facts about the triangles and in this example we only know two sides in each triangle.

Note that and are not straight lines and so we cannot use vertically opposite angles.

There is not enough information given. Although we can work out which angles are equal we are not given any sides as equal. All we know is that we have two isosceles triangles. Note how this question differs from part a). In part a) we were given equal sides in both triangles, in this question we are only given that sides in the same triangle are equal.

7.3 Quadrilaterals

Quadrilateral

A quadrilateral is a closed shape consisting of four straight line segments.

The interior angles of a quadrilateral add up to °.

Mathopenref has some useful simulations on different types of quadrilaterals. Clicking on any of the named quadrilaterals will take you to a page specific to that quadrilateral.

Parallelogram

Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Worked example 3: Properties of a parallelogram

is a parallelogram with and . Show that:

and

and

Connect to form and

Redraw the diagram and draw line .

Use properties of parallel lines to indicate all equal angles on the diagram

On your diagram mark all the equal angles.

Prove

In and :

Opposite sides of a parallelogram have equal length.

We have already shown and . Therefore,

Furthermore,

Therefore opposite angles of a parallelogram are equal.

Summary of the properties of a parallelogram:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

Worked example 4: Proving a quadrilateral is a parallelogram

Prove that if both pairs of opposite angles in a quadrilateral are equal, the quadrilateral is a parallelogram.

Find the relationship between and

In :

But these are co-interior angles between lines and . Therefore .

Find parallel lines

Similarly . These are co-interior angles between lines and . Therefore .

Both pairs of opposite sides of the quadrilateral are parallel, therefore is a parallelogram.

Proving a quadrilateral is a parallelogram

Prove that if both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Prove that if one pair of opposite sides of a quadrilateral are both equal and parallel, then the quadrilateral is a parallelogram.

A quadrilateral is a parallelogram if:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal.

Both pairs of opposite angles are equal.

The diagonals bisect each other.

One pair of opposite sides are both equal and parallel.

Exercise 7.3

is a parallelogram. and . and .

Find with reasons, two other angles equal to .

(alt s; ).

(s opp equal sides).

Therefore and are both equal to .

Write in terms of .

Calculate the value of .

Prove that the diagonals of parallelogram bisect one another at .

Hint: Use congruency.

First number each angle on the given diagram:

In and :

Therefore (AAS).

Now we know that and therefore is the mid-point of .

Similarly, in and :

Therefore (AAS).

Now we know that and therefore is the mid-point of .

Therefore the diagonals of a parallelogram bisect each other.

Rectangle

Rectangle

A rectangle is a parallelogram that has all four angles equal to .

A rectangle has all the properties of a parallelogram:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

It also has the following special property:

Worked example 5: Special property of a rectangle

is a rectangle. Prove that the diagonals are of equal length.

Connect to and to to form and

Use the definition of a rectangle to fill in on the diagram all equal angles and sides

Prove

In and :

The diagonals of a rectangle are of equal length.

Summary of the properties of a rectangle:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are of equal length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

Diagonals are equal in length.

All interior angles are equal to

Exercise 7.4

is a quadrilateral. Diagonals and intersect at . , , . Prove that:

is a parallelogram

(given)

bisects at

and (given)

bisects at

therefore quadrilateral is a parallelogram (diag of m)

is a rectangle

(given).

Therefore is a rectangle (diags of rectangle).

Rhombus

Rhombus

A rhombus is a parallelogram with all four sides of equal length.

A rhombus has all the properties of a parallelogram:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

It also has two special properties:

Worked example 6: Special properties of a rhombus

is a rhombus. Prove that:

the diagonals bisect each other perpendicularly;

the diagonals bisect the interior angles.

Use the definition of a rhombus to fill in on the diagram all equal angles and sides

Prove

We can further conclude that .

Therefore the diagonals bisect each other perpendicularly.

Use properties of congruent triangles to prove diagonals bisect interior angles

Therefore diagonal bisects . Similarly, we can show that also bisects ; and that diagonal bisects and .

We conclude that the diagonals of a rhombus bisect the interior angles.

To prove a parallelogram is a rhombus, we need to show any one of the following:

All sides are equal in length.

Diagonals intersect at right angles.

Diagonals bisect interior angles.

Summary of the properties of a rhombus:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

All sides are equal in length.

The diagonals bisect each other at

The diagonals bisect both pairs of opposite angles.

Square

Square

A square is a rhombus with all four interior angles equal to

OR

A square is a rectangle with all four sides equal in length.

A square has all the properties of a rhombus:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

All sides are equal in length.

The diagonals bisect each other at

The diagonals bisect both pairs of opposite angles.

It also has the following special properties:

All interior angles equal .

Diagonals are equal in length.

Diagonals bisect both pairs of interior opposite angles (i.e. all are ).

To prove a parallelogram is a square, we need to show either one of the following:

It is a rhombus (all four sides of equal length) with interior angles equal to .

It is a rectangle (interior angles equal to ).

Trapezium

Trapezium

A trapezium is a quadrilateral with one pair of opposite sides parallel.

In British English a trapezium is used to indicate a quadrilateral with one pair of opposite sides parallel while in American English a trapezium is a quadrilateral with no pairs of opposite sides parallel. We will use the British English definition of trapezium in this book.

In British English a trapezoid is used to indicate a quadrilateral with no pairs of opposite sides parallel while in American English a trapezoid is a quadrilateral with one pair of opposite sides parallel.

A trapezium is sometimes called a trapezoid.

Some examples of trapeziums are given below:

Kite

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal.

Worked example 7: Properties of a kite

is a kite with and . Prove that:

Diagonal bisects and

Prove

In and :

Therefore one pair of opposite angles are equal in kite .

Use properties of congruent triangles to prove bisects and

Therefore diagonal bisects and .

We conclude that the diagonal between the equal sides of a kite bisects the two interior angles and is an axis of symmetry.

Summary of the properties of a kite:

Diagonal between equal sides bisects the other diagonal.

One pair of opposite angles are equal (the angles between unequal sides).

Diagonal between equal sides bisects the interior angles and is an axis of symmetry.

Diagonals intersect at

Exercise 7.5

Use the sketch of quadrilateral to prove the diagonals of a kite are perpendicular to each other.

First number the angles:

In and :

In :

Next we note that:

Therefore the diagonals of a kite are perpendicular to each other.

Explain why quadrilateral is a kite. Write down all the properties of quadrilateral .

Quadrilateral is a kite because is has two pairs of adjacent sides that are equal in length.

Diagonal between equal sides bisects the other diagonal: .

One pair of opposite angles are equal: .

Diagonal between equal sides bisects the interior angles and is an axis of symmetry: .

Diagonals intersect at : .

This video provides a summary of the different types of quadrilaterals and their properties.

Video: 2G77

Relationships between the different quadrilaterals

Heather has drawn the following diagram to illustrate her understanding of the relationships between the different quadrilaterals. The following diagram summarises the different types of special quadrilaterals.

Explain her possible reasoning for structuring the diagram as shown.

Design your own diagram to show the relationships between the different quadrilaterals and write a short explanation of your design.

Exercise 7.6

The following shape is drawn to scale :

Give the most specific name for the shape.

We start by counting the number of sides. There are four sides in this figure and so it is either just a quadrilateral or one of the special types of quadrilateral.

Next we ask ourselves if there are any parallel lines in the figure. You can look at the figure to see if any of the lines look parallel or make a quick sketch of the image and see if any pairs of opposite lines meet at a point.

Both pairs of opposite sides are parallel. This means that the figure can only be one of the following: parallelogram, rectangle, rhombus or square.

Next we ask ourselves if all the interior angles are 90°. All the interior angles are 90° and so this must be a square or a rectangle. Finally we check to see if all the sides are equal in length. In this figure the sides are not equal in length and so it is a rectangle.

Therefore this is a rectangle.

The shape is also a parallelogram and a quadrilateral. This question, however, asked for the most specific name for the shape.

The following shape is drawn to scale :

Give the most specific name for the shape.

We start by counting the number of sides. There are four sides in this figure and so it is either just a quadrilateral or one of the special types of quadrilateral.

Next we ask ourselves if there are any parallel lines in the figure. You can look at the figure to see if any of the lines look parallel or make a quick sketch of the image and see if any pairs of opposite lines meet at a point.

Both pairs of opposite sides are parallel. This means that the figure can only be one of the following: parallelogram, rectangle, rhombus or square.

Next we ask ourselves if all the interior angles are 90°. All the interior angles are not 90° and so this must be a parallelogram or a rhombus. Finally we check to see if all the sides are equal in length. In this figure the sides are equal in length and so it is a rhombus.

Therefore this is a rhombus.

The shape is also a parallelogram and a quadrilateral. This question, however, asked for the most specific name for the shape.

Based on the shape that you see list the all the names of the shape. The figure is drawn to scale

Both pairs of opposite sides are not parallel. This means that the figure can only be some combination of the following: trapezium, kite, or quadrilateral.

The shape is definitely a quadrilateral because it has four sides. It does not have any special properties: it does not have parallel sides, or right angles, or sides which are equal in length. Therefore it is only a quadrilateral.

Based on the shape that you see list the all the names of the shape. The figure is drawn to scale

Both pairs of opposite sides are not parallel. This means that the figure can only be some combination of the following: trapezium, kite, or quadrilateral.

The shape is definitely a quadrilateral because it has four sides. It is also a kite because it has two pairs of adjacent sides which are the same lengths. It cannot be a square or a rectangle because it does not have right angles. It cannot be a parallelogram or a trapezium because it does not have any parallel sides. And it is not a rhombus because the four sides are not all the same length.

Therefore the correct answer is: kite and quadrilateral.

Based on the shape that you see list the all the names of the shape. The figure is drawn to scale

Both pairs of opposite sides are parallel. That means that this shape can belong to one or more of these groups: square, rhombus, rectangle or parallelogram.

The given shape is a square. However, it is also a rectangle. A square is also a parallelogram, because it has parallel sides; and it is a rhombus as well, it just happens to have right angles. A square is also a kite, a trapezium and of course a quadrilateral.

Therefore the correct answer is: square, rectangle, rhombus, parallelogram, kite, trapezium and quadrilateral.

Find the area of if

Construct such that

is a rhombus (diagonals bisect at right angles)

Since is a rhombus . We constructed such that . Therefore .

In and :

Therefore (RHS)

Therefore and so is a rectangle (both pairs of opposite sides equal in length and all interior angles are 90°).

We are given the length of and . Since is right-angled we can use the theorem of Pythagoras to find the length of :

We also know that and so .

Therefore the area of rectangle is:

We are almost there. We now need to calculate the area of triangle and subtract this from the area of the rectangle to get the area of .

The area of triangle is:

Therefore the area of is .

7.4 The mid-point theorem

Proving the mid-point theorem

Draw a large scalene triangle on a sheet of paper.

Name the vertices , and . Find the mid-points ( and ) of two sides and connect them.

Cut out and cut along line .

Place on quadrilateral with vertex on vertex . Write down your observations.

Shift to place vertex on vertex . Write down your observations.

What do you notice about the lengths and ?

Make a conjecture regarding the line joining the mid-point of two sides of a triangle.

Worked example 8: Mid-point theorem

Prove that the line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

Extend to so that and join

Prove is a parallelogram

In and :

But these are alternate interior angles, therefore

Therefore .

We conclude that the line joining the two mid-points of two sides of a triangle is parallel to the third side.

Use properties of parallelogram to prove that

We conclude that the line joining the mid-point of two sides of a triangle is equal to half the length of the third side.

Converse

The converse of this theorem states: If a line is drawn through the mid-point of a side of a triangle parallel to the second side, it will bisect the third side.

You can use GeoGebra to show that the converse of the mid-point theorem is true.

Exercise 7.7

Points and are the mid-points on lines and . Study carefully. Identify the third side of this triangle, using the information as shown, together with what you know about the mid-point theorem. Name the third side by its endpoints, e.g., .

The red line, or , indicates the third side of the triangle. According to the mid-point theorem, the line joining the mid-points of two sides of a triangle is parallel to the third side of the triangle.

Points and are the mid-points on lines and . Study carefully. Identify the third side of this triangle, using the information as shown, together with what you know about the mid-point theorem. Name the third side by its endpoints, e.g., .

The red line, or , indicates the third side of the triangle. According to the mid-point theorem, the line joining the mid-points of two sides of a triangle is parallel to the third side of the triangle.

Points and are given on the lines and . Study the triangle carefully, then identify and name the parallel lines.

The lines and are parallel according to the mid-point theorem because is bisecting the lines and .

Points and are given on the lines and . Study the triangle carefully, then identify and name the parallel lines.

The lines and are not parallel according to the mid-point theorem because line does not bisect and . Therefore there are no parallel lines in the triangle.

The figure below shows a large triangle with vertices , and , and a smaller triangle with vertices at , and . Point is the mid-point of and point is the mid-point of .

Three angles are given: , and ; determine the value of .

The two triangles in this question are similar triangles. Complete the following statement correctly by giving the three vertices in the correct order (there is only one correct answer).

Angle corresponds to angle ; angle corresponds to angle ; and angle corresponds to angle . Therefore, .

The figure below shows a large triangle with vertices , and , and a smaller triangle with vertices at , and . Point is the mid-point of and point is the mid-point of .

Three angles are given: , , and ; determine the value of .

The two triangles in this question are similar triangles. Complete the following statement correctly by giving the three vertices in the correct order (there is only one correct answer).

Angle corresponds to angle ; angle corresponds to angle ; and angle corresponds to angle . Therefore, .

Consider the triangle in the diagram below. There is a line crossing through a large triangle. Notice that some lines in the figure are marked as equal to each other. One side of the triangle has a given length of 3. Some information is also given about the lengths of other lines along the edges of the triangle.

Determine the value of .

From the mid-point theorem we know:

Consider the triangle in the diagram below. There is a line crossing through a large triangle. Notice that some lines in the figure are marked as equal to each other. One side of the triangle has a given length of 6.

Determine the value of .

From the mid-point theorem we know:

In the figure below, , as labelled. Furthermore, the following lengths and angles are given: ; ; ; and . The figure is drawn to scale.

Determine the length of .

is the mid-point of and is the mid-point of (, also it is given that ).

In the figure below, , as labelled. Furthermore, the following lengths and angles are given: ; ; ; and . The figure is drawn to scale.

Determine the length of .

is the mid-point of and is the mid-point of (, also it is given that ).

Find and in the following:

From the mid-point theorem we know:

From the mid-point theorem we know:

We can use the theorem of Pythagoras to find :

From the mid-point theorem we know:

From the mid-point theorem we know:

To find we note the following:

(sum of s in ).

From the mid-point theorem we also know that .

Therefore (corresp s; ).

The final answer is: and .

In the following diagram and .

From the mid-point theorem we know that . Therefore (corresp s; ).

Therefore (sum of s in ).

To find we note that and , therefore . Similarly .

We can use the theorem of Pythagoras to find :

Therefore: and .

Show that is the mid-point of and that .

We are given that .

We are also given that , therefore ( and are equal, corresponding angles).

Therefore is the mid-point of (converse of mid-point theorem).

Similarly we can show that is the mid-point of .

We also know that ( and parallel lines are a constant distance apart).

But ( is the mid-point of ), therefore .

In the diagram below, is the mid-point of and is the mid-point of . The segment inside of the large triangle is labelled with a length of .

Calculate the value of in terms of .

Use the mid-point theorem to fill in known information on the diagram:

Remember that the mid-point theorem tells us that the segments and have a ratio of ( is twice as long as ).

The final answer is (twice as long as ).

You are now told that has a length of . What is the value of ? Give your answer as a fraction.

In the diagram below, is the mid-point of and is the mid-point of . One side of the triangle has a given length of .

Find the value of in terms of .

Use the mid-point theorem to fill in known information on the diagram:

Remember that the mid-point theorem tells us that the segments and have a ratio of ( is twice as long as ).

You are now told that has a length of . What is the value of ?

The figure below shows crossed by . Points and bisect their respective sides of the triangle.

The angles and are given; determine the value of in terms of .

We note the following from the mid-point theorem:

Also

In terms of , the answer is: .

You are now told that has a measure of . Calculate for the value of .

The figure below shows crossed by . Points and bisect their respective sides of the triangle.

Given the angles and , determine the value of in terms of .

Using the mid-point theorem we can add the following information to the diagram:

Also:

In terms of , the answer is: .

You are now told that has a measure of . Solve for the value of . Give your answer as a fraction.

7.5 Chapter summary

Presentation: 2G86

A quadrilateral is a closed shape consisting of four straight line segments.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

A rectangle is a parallelogram that has all four angles equal to

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

The diagonals bisect each other.

The diagonals are equal in length.

All interior angles are equal to .

A rhombus is a parallelogram that has all four sides equal in length.

Both pairs of opposite sides are parallel.

All sides are equal in length.

Both pairs of opposite angles are equal.

The diagonals bisect each other at .

The diagonals of a rhombus bisect both pairs of opposite angles.

A square is a rhombus that has all four interior angles equal to .

Both pairs of opposite sides are parallel.

The diagonals bisect each other at .

All interior angles are equal to .

The diagonals are equal in length.

The diagonals bisect both pairs of interior opposite angles (i.e. all are

A trapezium is a quadrilateral with one pair of opposite sides parallel.

A kite is a quadrilateral with two pairs of adjacent sides equal.

One pair of opposite angles are equal (the angles are between unequal sides).

The diagonal between equal sides bisects the other diagonal.

The diagonal between equal sides bisects the interior angles.

The diagonals intersect at .

The mid-point theorem states that the line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

Exercise 7.8

Identify the types of angles shown below:

straight angle

obtuse angle

acute angle

right angle

Reflex angle

An angle of °

obtuse angle

An angle of °

straight angle

An angle of °

reflex angle

Assess whether the following statements are true or false. If the statement is false, explain why:

A trapezium is a quadrilateral with two pairs of opposite sides that are parallel.

False, a trapezium only has one pair of opposite parallel sides.

Both diagonals of a parallelogram bisect each other.

True

A rectangle is a parallelogram that has all interior angles equal to °.

True

Two adjacent sides of a rhombus have different lengths.

False, two adjacent sides of a rhombus are equal in length.

The diagonals of a kite intersect at right angles.

True

All squares are parallelograms.

True

A rhombus is a kite with a pair of equal, opposite sides.

True

The diagonals of a parallelogram are axes of symmetry.

True

The diagonals of a rhombus are equal in length.

False, the diagonals of a rhombus are not equal in length.

Both diagonals of a kite bisect the interior angles.

False, only one diagonal of a kite bisects one pair of interior angles.

Find all pairs of parallel lines in the following figures, giving reasons in each case.

(alt s equal)

Using the sum of angles on a straight line we can state the following:

not (corresp s not equal).

(corresp s equal).

(s on a str line).

Therefore (corresp s equal).

And (alt s equal).

The pairs of parallel lines are and .

Find angles , , and in each case, giving reasons:

Therefore .

Therefore .

Therefore .

Find each of the unknown angles marked in the figure below. Find a reason that leads to the answer in a single step.

and are alternate interior angles on transversal . , therefore they must be equal in size.

Therefore .

corresponds to (matches) ; and corresponding angles are equal in size since .

Therefore: .

(s on str line):

and are vertically opposite angles and vertically opposite angles have the same measure (equal sizes).

Therefore: .

Based on the results for the angles above, is ?

If is parallel to , then the following things must all be true:

(corresponding angles)

(alternate interior angles)

(co-interior angles)

All the above is true, therefore the lines are parallel.

Given the following diagrams:

Diagram A

Diagram B

Which diagram correctly gives a pair of similar triangles?

We look at the side labels. In diagram A we note that the three pairs of corresponding sides are in different proportions. In diagram B we note the three pairs of corresponding sides are in proportion.

Therefore diagram B gives a pair of triangles that are similar.

Given the following diagrams:

Diagram A

Diagram B

Which diagram correctly gives a pair of similar triangles?

Diagram A shows a pair of triangles with all pairs of corresponding angles equal (the same three angle markers are shown in both triangles). Diagram B shows a pair of triangles with different angles in each triangle. All six angles are different and there are no pairs of corresponding angles that are equal.

Therefore diagram A gives a pair of triangles that are similar.

Have a look at the following triangles, which are drawn to scale:

Are the triangles congruent? If so state the reason and use correct notation to state that they are congruent.

We are given one angle that is equal. We are not given any equal sides (we do not know if ). To determine if two triangles are congruent we need to have three pieces of information (recall that the reasons for congruent triangles are: SSS, SAS, AAS and RHS). Therefore we cannot state whether or not the triangles are congruent.

Therefore, there is not enough information to determine if the two triangles are congruent.

Have a look at the following triangles, which are drawn to scale:

Are the triangles congruent? If so state the reason and use correct notation to state that they are congruent.

The sides of both triangles are labelled with , and . This means that there are three pairs of corresponding and equal sides.

Therefore, these two triangles are congruent (), reason: SSS.

Say which of the following pairs of triangles are congruent with reasons.

We are given , and .

Therefore by SSS.

We are given , and .

Therefore by RHS.

We are given , and .

Therefore by AAS.

We are given , and . But is not the included angle between sides and .

Therefore .

Using the theorem of Pythagoras, calculate the length :

First find :

Now we note that and then we find :

Calculate and in the diagrams below:

(sum of s in ).

(sum of s in ).

We can find using the theorem of Pythagoras:

We note that the triangles are similar by AAA. Therefore the sides must be in proportion. Therefore is:

Therefore and .

This is an isosceles triangle so .

Therefore (sum of s in ).

This is an isosceles triangle so .

The two triangles are similar by AAA. Therefore the sides are in proportion.

Therefore and .

Consider the diagram below. Is ? Give reasons for your answer.

All three pairs of sides are in proportion, .

Explain why is similar to and calculate the values of and .

However (sum of s in ). Therefore .

Similarly .

But so and so . Therefore .

Therefore (AAA).

Now we can use the fact that the sides are in proportion to find :

Therefore and .

The following shape is drawn to scale:

Give the most specific name for the shape.

We start by counting the number of sides. There are four sides in this figure and so it is either just a quadrilateral or one of the special types of quadrilateral.

Next we ask ourselves if there are any parallel lines in the figure. You can look at the figure to see if any of the lines look parallel or make a quick sketch of the image and see if any pairs of opposite lines meet at a point.

Both pairs of opposite sides are not parallel. This means that the figure can only be one of the following: trapezium, kite or quadrilateral.

Next we ask ourselves if one of the pairs of opposite sides is parallel, while the other is not. Neither of the two pairs of opposite sides is parallel so we must now look to see if both pairs of adjacent sides are equal in length. Both pairs of adjacent sides are equal in length. So this is a kite.

Therefore this is a kite.

Based on the shape that you see list the all the names of the shape. The figure is drawn to scale.

Both pairs of opposite sides are parallel. That means that this shape can belong to one or more of these groups: square, rhombus, rectangle, and/or parallelogram.

The shape shown is a rhombus. It is certainly a quadrilateral (because it has four sides). It is also a parallelogram, because the opposite sides are parallel to each other. The rhombus is not a rectangle or a square because it does not have right angles. However, the rhombus is a kite, because it has two pairs of adjacent sides which are equal in length. And finally, it is a trapezium because it has a pair of opposite sides which are parallel.

Therefore the correct answer is: rhombus, parallelogram, kite, trapezium and quadrilateral.

is a rhombus. ; . Determine the value of .

(diagonals of a rhombus bisect at right angles)

In the diagram below, .

Name:

rectangles

, and

parallelograms

, , and

trapeziums

and

rhombi

and

Points and are the mid-points on lines and . Study carefully. Identify the third side of this triangle, using the information as shown, together with what you know about the mid-point theorem. (Name the third side by its endpoints, e.g., .)

The red line, or , indicates the third side of the triangle. According to the mid-point theorem, the line joining the mid-points of two sides of a triangle is parallel to the third side of the triangle.

Points and are given on the segments and . Study the triangle carefully, then identify and name the parallel line segments.

The line segments and are parallel according to the mid-point theorem because segment is bisecting the line segments and .

The figure below shows a large triangle with vertices , and , and a smaller triangle with vertices at , and . Point is the mid-point of and point is the mid-point of .

The angles and are given; determine the value of .

The two triangles in this question are similar triangles. Complete the following statement correctly by giving the three vertices in the correct order (there is only one correct answer).

Angle corresponds to angle ; angle corresponds to angle ; and angle corresponds to angle . Therefore, .

The figure below shows a large triangle with vertices , and , and a smaller triangle with vertices at , and . Point is the mid-point of and point is the mid-point of .

With the two angles given, and , determine the value of .

The two triangles in this question are similar triangles. Complete the following statement correctly by giving the three vertices in the correct order (there is only one correct answer).

Angle corresponds to angle ; angle corresponds to angle ; and angle corresponds to angle . Therefore, .

Consider the triangle in the diagram below. There is a line segment crossing through a large triangle. Notice that some segments in the figure are marked as equal to each other. One side of the triangle has a given length of 10. Some information is also given about the lengths of other segments along the edges of the triangle.

From the mid-point theorem we know:

In the figure below, , as labelled. Furthermore, the following lengths and angles are given: ; ; ; and . The figure is drawn to scale.

Calculate the length of .

We are given that . The length of is 5 and the length of is 10, therefore .

Therefore we know from the mid-point theorem that is the mid-point of and is the mid-point of .

Therefore .

The figure below shows triangle with the smaller triangle sitting inside of it. Furthermore, the following lengths and angles are given: ; . The figure is drawn to scale.

Find the length of .

You can see in the figure that the segment is not parallel to . This means that the mid-point theorem cannot apply to this triangle. There are no other options to use either: this question cannot be solved. There is no solution.

In the diagram below, is the mid-point of and is the mid-point of . One side of the triangle has a given length of .

Determine the value of in terms of .

Fill in information on the diagram using the mid-point theorem:

Remember that the mid-point theorem tells us that the segments and have a ratio of ( is twice as long as ).

The final answer is (half the size as ).

You are now told that has a length of . What is the value of ?

The final answer is .

The figure below shows crossed by . Points and bisect their respective sides of the triangle.

With the two angles given, and , determine the value of in terms of .

Redraw the diagram and fill in the known information using the mid-point theorem:

(s on str line).

(corresp s; ).

Therefore:

You are now told that has a measure of . Determine for the value of . Give your answer as an exact fractional value.

The figure below shows crossed by . Points and bisect their respective sides of the triangle.

The angles and are given in the large triangle; determine the value of in terms of .

Redraw the diagram and fill in known information using the mid-point theorem:

(corresp s; ).

You are now told that has a measure of . Solve for the value of . Give your answer as an exact fractional value.

Calculate and :

In and :

Therefore (AAA).

Therefore the sides are in proportion.

Therefore: and .

and are equilateral triangles. Prove that is a rhombus.

We are given two equilateral triangles, therefore in : and in : .

But is a common side and so .

Also in each triangle all the interior angles are equal to 60°. Therefore and . Therefore and (alt. int. 's equal).

is a rhombus (all sides are equal in length, both pairs of opposite sides parallel).

is a quadrilateral with and diagonals that intersect at such that . Prove that:

We need to show that one of or or or is equal to 90°.

We have already proved that and that (using congruent triangles).

We also note that .

Next we note that:

Now we note that:

Using the figure below, show that the sum of the three angles in a triangle is 180°. Line is parallel to .

(given).

(alt s; ).

(alt s; ).

(s on str line).

And we have shown that and therefore we can replace with and with to get:

.

Therefore the angles in a triangle add up to .

is an isosceles triangle with . is the mid-point of , is the mid-point of and is the mid-point of .

Prove is also isosceles.

What type of quadrilateral is ? Motivate your answer.

is a rhombus. It is a parallelogram since and (from the mid-point theorem) with four equal sides: (given and proved above).

If calculate, with reasons, the size of .

is a parallelogram. . Prove that .

In the diagram below, , and are the mid-points of , and respectively. .

Prove that is a parallelogram.

Prove that .

Chapter 8: Analytical geometry

This chapter covers representing geometric figures on the Cartesian co-ordinate system. Also covered are the distance formula, gradient of a line and mid-point of a line.

Distance formulae, gradient of a line and mid-point of a line should first be derived and then applied to solving problems.

Integrate Euclidean geometry knowledge with analytical geometry. It may be helpful to have learners write down the properties of the special quadrilaterals and keep this handy while working through analytical geometry.

Emphasise the value and importance of making sketches.

Emphasise the importance of writing coordinates consistently for the distance formula and gradient.

This chapter also draws strongly on the equation of a straight line. Ensure learners are comfortable working with the equation of a straight line.

mathopenref.com has many interactive elements that you can use while teaching analytical geometry.

Analytical geometry is the study of geometric properties, relationships and measurement of points, lines and angles in the Cartesian plane. Geometrical shapes are defined using a coordinate system and algebraic principles. Some consider the introduction of analytical geometry, also called coordinate or Cartesian geometry, to be the beginning of modern mathematics.

Figure 8.1: The motion of a projectile can be plotted on the Cartesian plane. Animators use this information to help them create animations.

8.1 Drawing figures on the Cartesian plane

If we are given the coordinates of the vertices of a figure, we can draw the figure on the Cartesian plane. For example, quadrilateral with coordinates , , and .

We use a semi-colon (;) to separate the and values but the internationally accepted method is to use a comma (,). If a comma is used than it becomes unclear as to whether the comma is separating the and values or one of the values is a decimal. For example the point is ambiguous. Is the value or is the value ?

You might also see coordinates written as .

The order of the letters for naming a figure is important. It indicates the order in which points must be joined: to , to , to and back to . So the above quadrilateral can be referred to as quadrilateral or or . However it is conventional to write the letters in alphabetical order and so we only refer to the quadrilateral as .

You can use an online tool to help you when plotting points on the Cartesian plane. Click here to try this one on mathsisfun.com.

Exercise 8.1

You are given the following diagram, with various points shown:

Find the coordinates of point .

For this question, we are only interested in point . From the graph we can read off the and values.

Point has the following coordinates: .

You are given the following diagram, with various points shown:

Find the coordinates of all the labelled points.

From the graph we can read off the and values for each point.

, , , and .

You are given the following diagram, with various points shown:

Which point lies at the coordinates ?

For this question, we are must find point .

On the graph we can trace the and values to find which point lies at the coordinates .

Doing so we find that point lies at the coordinates .

You are given the following diagram, with various points shown:

Which point lies at the coordinates ?

For this question, we are must find point .

On the graph we can trace the and values to find which point lies at the coordinates .

Doing so we find that point lies at the coordinates .

You are given the following diagram, with 4 shapes drawn.

All the shapes are identical, but each shape uses a different naming convention:

Which shape uses the correct naming convention?

We recall that the correct naming convention for a shape is in alphabetical order, either clockwise or anti-clockwise around the shape.

From the diagram, we can see that only shape Z sticks to this naming convention.

You are given the following diagram, with 4 shapes drawn.

All the shapes are identical, but each shape uses a different naming convention:

Which shape uses the correct naming convention?

We recall that the correct naming convention for a shape is in alphabetical order, either clockwise or anti-clockwise around the shape.

From the diagram, we can see that only shape Z sticks to this naming convention.

8.2 Distance between two points

A point is a simple geometric object having location as its only property.

Point

A point is an ordered pair of numbers written as .

Distance

Distance is a measure of the length between two points.

Distance between two points

Points , and are given.

Can we assume that ? If so, why?

Apply the theorem of Pythagoras in to find the length of .

To derive a general formula for the distance between two points and we use the theorem of Pythagoras.

And:

Therefore:

Therefore to calculate the distance between any two points, and , we use:

Note that .

The following video gives two examples of working with the distance formula and shows how to determine the distance formula.

Video: 2GBS

Worked example 1: Using the distance formula

Find the distance between and .

Draw a sketch

Assign values to and

Let the coordinates of be and the coordinates of be .

Write down the distance formula

Substitute values

Write the final answer

The distance between and is units.

Worked example 2: Using the distance formula

Given , and . Find .

Draw a sketch

Note that we expect two possible values for . This is because the distance formula includes the term which results in a quadratic equation when we substitute in the coordinates.

Assign values to and

Let the coordinates of be and the coordinates of be .

Write down the distance formula

Substitute values and solve for

Check both values for

Check :

Check :

Write the final answer

is or .

Therefore or .

Drawing a sketch helps with your calculation and makes it easier to check if your answer is correct.

Exercise 8.2

You are given the following diagram:

Calculate the length of line , correct to 2 decimal places.

First we recall the equation for distance:

You are given the following diagram:

Calculate the length of line , correct to 2 decimal places.

First we recall the equation for distance:

The following picture shows two points on the Cartesian plane, and .

The distance between the points is . Calculate the missing coordinate of point .

First we recall the equation for distance:

Now we re-arrange, and solve for the value of :

We now have a choice between 2 values for . From the diagram we can see that the appropriate value for this question is .

Note that in this case we can use the diagram to check that our answer is valid but we can also calculate the distance of line using our answer.

The following picture shows two points on the Cartesian plane, and .

The line has a length of . Calculate the missing coordinate of point . Round your answer to one decimal place.

First we recall the equation for distance:

Now we re-arrange, and solve for the value of :

We now have a choice between 2 values for . From the diagram we can see that the appropriate value for this question is .

Note that in this case we can use the diagram to check that our answer is valid but we can also calculate the distance of line using our answer.

Find the length of for each of the following. Leave your answer in surd form.

and

and

and

The length of . Find the missing coordinate if:

and .

Therefore or .

Check solution for :

Solution is valid.

Check solution for :

Solution is valid.

and .

Therefore or .

Check solution for :

Solution is valid.

Check solution for :

Solution is valid.

If the distance between and is 10 units, find the possible values of .

Check :

Solution is valid.

Check :

Solution is valid.

Therefore or .

8.3 Gradient of a line

Gradient

The gradient of a line is determined by the ratio of vertical change to horizontal change.

Gradient () describes the slope or steepness of the line joining two points. In the figure below, line is the least steep and line is the steepest.

To derive the formula for gradient, we consider any right-angled triangle formed from and with hypotenuse as shown in the diagram below. The gradient is determined by the ratio of the length of the vertical side of the triangle to the length of the horizontal side of the triangle. The length of the vertical side of the triangle is the difference in -values of points and . The length of the horizontal side of the triangle is the difference in -values of points and .

Therefore, gradient is determined using the following formula:

Gradient or

Remember to be consistent:

To learn more about determining the gradient you can watch the following video.

Video: 2GC6

Worked example 3: Gradient between two points

Find the gradient of the line between points and .

Draw a sketch

Assign values to and

Let the coordinates of be and the coordinates of be .

Write down the formula for gradient

Substitute known values

Write the final answer

The gradient of

Worked example 4: Gradient between two points

Given and , with , find .

Draw a sketch

Assign values to and

Let the coordinates of be and the coordinates of be

Write down the formula for gradient

Substitute values and solve for x

Write the final answer

The coordinates of are .

Therefore .

Exercise 8.3

Find the gradient of if:

and

Let the coordinates of be and the coordinates of be

and

Let the coordinates of be and the coordinates of be

and

Let the coordinates of be and the coordinates of be

You are given the following diagram:

Calculate the gradient () of line .

Let the coordinates of be and the coordinates of be

Therefore the gradient of the line is .

You are given the following diagram:

Calculate the gradient () of line .

Let the coordinates of be and the coordinates of be

Therefore the gradient of the line is .

If the gradient of , find given:

and .

Let the coordinates of be and the coordinates of be

and .

Let the coordinates of be and the coordinates of be

You are given the following diagram:

You are also told that line has a gradient () of .

Calculate the missing co-ordinate of point .

Let the coordinates of be and the coordinates of be

You are given the following diagram:

You are also told that line has a gradient () of .

Calculate the missing co-ordinate of point .

Let the coordinates of be and the coordinates of be

Straight lines

Straight line

A straight line is a set of points with a constant gradient between any two of the points.

Consider the diagram below with points , and .

We have and

The general formula for finding the equation of a straight line is where is any point on the line.

This formula can also be written as .

The standard form of the straight line equation is where is the gradient and is the -intercept.

The worked example below shows one method of determining the equation of a straight line. The equation of a straight line can also be determined by first finding the gradient and then substituting one of the given points into .

Worked example 5: Finding the equation of a straight line

Find the equation of the straight line through and .

Draw a sketch

Assign values

Let the coordinates of be and the coordinates of be .

Write down the general formula of the line

Substitute values and make the subject of the equation

Write the final answer

The equation of the straight line is .

Parallel and perpendicular lines

Two lines that run parallel to each other are always the same distance apart and have equal gradients.

If two lines intersect perpendicularly, then the product of their gradients is equal to .

If line , then . Perpendicular lines have gradients that are the negative inverses of each other.

The following video shows some examples of calculating the slope of a line and determining if two lines are perpendicular or parallel.

Video: 2GCJ

Worked example 6: Parallel lines

Prove that line with and is parallel to line with equation .

Draw a sketch

(Be careful - some lines may look parallel but are not!)

Write down the formula for gradient

Substitute values to find the gradient for line

Check that the equation of is in the standard form

Write the final answer

therefore line is parallel to line .

Worked example 7: Perpendicular lines

Line is perpendicular to line . Find given , , and .

Draw a sketch

Write down the relationship between the gradients of the perpendicular lines

Substitute values and solve for

Write the final answer

Therefore the coordinates of are .

Horizontal and vertical lines

A line that runs parallel to the -axis is called a horizontal line and has a gradient of zero. This is because there is no vertical change:

A line that runs parallel to the -axis is called a vertical line and its gradient is undefined. This is because there is no horizontal change:

Points on a line

A straight line is a set of points with a constant gradient between any of the two points. There are two methods to prove that points lie on the same line: the gradient method and a method using the distance formula.

If two points lie on the same line then the two points are said to be collinear.

Worked example 8: Points on a line

Prove that , and are on a straight line.

Draw a sketch

Calculate two gradients between any of the three points

and

OR

and

Explain your answer

Therefore the points , and are on a straight line.

To prove that three points are on a straight line using the distance formula, we must calculate the distances between each pair of points and then prove that the sum of the two smaller distances equals the longest distance.

Worked example 9: Points on a straight line

Prove that , and are on a straight line.

Draw a sketch

Calculate the three distances , and

Find the sum of the two shorter distances

Explain your answer

therefore points , and lie on the same straight line.

Exercise 8.4

Determine whether and are parallel, perpendicular or neither if:

, , ,

We need to calculate the gradients of lines and . Then we can compare the gradients to determine if the lines are parallel (the gradients are equal), perpendicular (the gradients are the negative inverses of each other) or neither.

And:

Therefore:

Lines and are parallel.

, , ,

We need to calculate the gradients of lines and . Then we can compare the gradients to determine if the lines are parallel (the gradients are equal), perpendicular (the gradients are the negative inverses of each other) or neither.

And:

So . Therefore is not parallel to .

And . Therefore and are not perpendicular.

Lines and are neither parallel nor perpendicular.

, , ,

We need to calculate the gradients of lines and . Then we can compare the gradients to determine if the lines are parallel (the gradients are equal), perpendicular (the gradients are the negative inverses of each other) or neither.

And:

So . Therefore is not parallel to .

And . Therefore and are perpendicular.

Determine whether the following points lie on the same straight line:

, ,

And,

So and is a common point. Therefore , and are collinear (they lie on the same line).

, ,

And,

So and is a common point. Therefore , and are collinear (they lie on the same line).

, ,

And,

So . Therefore , and are not collinear (they do not lie on the same line).

You are given the following diagram:

Calculate the equation of the line .

To calculate the equation of the straight line, we first calculate the gradient () of the line :

Secondly, we calculate the value of the -intercept () of the line . We do this by substituting either point or into the general form for a straight line. We will use point .

Therefore, the equation of the line is as follows:

You are given the following diagram:

Calculate the equation of the line .

To calculate the equation of the straight line, we first calculate the gradient () of the line :

Secondly, we calculate the value of the -intercept () of the line . We do this by substituting either point or into the general form for a straight line. We will use point .

Therefore, the equation of the line is as follows:

Points , and lie on a straight line. Find the value of .

Since the three points lie on a straight line we can use the fact that the gradient of is equal to the gradient of to find .

The gradient of is:

And the gradient of in terms of is:

Now we let and solve for :

Line with and has a gradient of . Find .

You are given the following diagram:

You are also told that line runs parallel to the following line: . Point is at .

Find the equation of the line .

We are told that line is parallel to , so the gradient of line is equal to the gradient of . The gradient of is 1.

Now we can use point and the gradient of the line to find the -intercept of the line:

Therefore, the equation of the line is: .

You are given the following diagram:

You are also told that line runs parallel to the following line: . Point is at .

Find the equation of the line .

We are told that line is parallel to , so the gradient of line is equal to the gradient of . The gradient of is .

Now we can use point and the gradient of the line to find the -intercept of the line:

Therefore, the equation of the line is: .

Given line which runs parallel to . Points and are also given.

Calculate the missing co-ordinate of point .

We are told that line is parallel to , so the gradient of line is equal to the gradient of . The gradient of is .

Now we can use point and the gradient of the line to find the -intercept of the line:

Therefore, the equation of the line is: .

Now we can substitute point into the equation of line to solve for :

Therefore the coordinates of point are .

Given line which runs parallel to . Points and are also given.

Calculate the missing co-ordinate of point .

We are told that line is parallel to , so the gradient of line is equal to the gradient of . The gradient of is .

Now we can use point and the gradient of the line to find the -intercept of the line:

Therefore, the equation of the line is: .

Now we can substitute point into the equation of line to solve for :

Therefore the coordinates of point are .

The graph here shows line . The blue dashed line is perpendicular to .

The equation of the blue dashed line is . Point is at .

Determine the equation of line .

The general form of a straight line is: .

Line is perpendicular to the blue dashed line and so .

Now we can substitute in the coordinates of point to find the -intercept:

Therefore, the equation of the line is: .

The graph here shows line . The blue dashed line is perpendicular to .

The equation of the blue dashed line is . Point is at .

Determine the equation of line .

The general form of a straight line is: .

Line is perpendicular to the blue dashed line and so .

Now we can substitute in the coordinates of point to find the -intercept:

Therefore, the equation of the line is: .

Given line which runs perpendicular to line with equation . Points and are also given.

Calculate the missing co-ordinate of point .

The general form of a straight line is: .

Line is perpendicular to line and so .

Now we can substitute point into the equation to find the -intercept:

Next we can substitute in point to find the missing coordinate:

Therefore the missing coordinate is .

Given line which runs perpendicular to line with equation . Points and are also given.

Calculate the missing co-ordinate of point .

The general form of a straight line is: .

Line is perpendicular to line and so .

Now we can substitute point into the equation to find the -intercept:

Next we can substitute in point to find the missing coordinate:

Therefore the missing coordinate is .

You are given the following diagram:

You are also told that line has the following equation: .

Calculate the missing co-ordinate of point .

We substitute the known value for point into the equation and solve for the unknown value:

You are given the following diagram:

You are also told that line has the following equation: .

Calculate the missing coordinate of point .

We substitute the known value for point into the equation and solve for the unknown value:

is the point and is the point . is perpendicular to line with equation . Find the value of .

Line is perpendicular to line and so .

Therefore:

The points , and are given. Determine the value of if , and are collinear.

We are told that , and are collinear, therefore .

Refer to the diagram below:

Show that is right-angled. Show your working.

To show that is right angled we need to show that or that or . We can do this by calculating the gradients of , and and then seeing if any of these gradients is the negative inverse of any of the other two gradients.

Now we note that:

Find the area of .

This is a right-angled triangle (with right-angle ) and so the perpendicular height is the length of one of the sides. We will use side as the perpendicular height and side as the base.

Note that we cannot use side for either the base or the height as we would then need to construct a new perpendicular height from to and calculate the length of that line.

The length of is:

The length of is:

Therefore the area of is:

The points , and are given.

Prove that triangle is a right-angled triangle.

We first draw a sketch:

To show that is right angled we need to show that or that or . We can do this by calculating the gradients of , and and then seeing if any of these gradients is the negative inverse of any of the other two gradients.

Now we note that:

Find the coordinates of , if is a parallelogram.

A parallelogram has both sides equal in length and parallel. Therefore and . Also and .

Let the coordinates of be .

Since , .

From the previous question we know that . Therefore the gradient of is:

Since , .

From the previous question we know that . Therefore the gradient of is:

Now we have two equations with two unknowns. We can equate the two equations and solve for :

Now we can solve for :

Therefore the coordinates of are .

Find the equation of a line parallel to the line , which passes through the point .

We know from the first question that the gradient of line is 2. We also know that point is at .

The line parallel to will have the same gradient as so we can write the equation of this line as: .

Now we can use point to find the -intercept of the line:

Therefore the equation of the line parallel to and passing through is .

8.4 Mid-point of a line

Finding the mid-point of a line

On graph paper, accurately plot the points and and draw the line .

Fold the piece of paper so that point is exactly on top of point .

Where the folded line intersects with line , label point

Count the blocks and find the exact position of .

Write down the coordinates of .

To calculate the coordinates of the mid-point of any line between the points and , we use the following formulae:

From this we obtain the mid-point of a line:

Mid-point

This video shows some examples of finding the mid-point of a line.

Video: 2GDF

Worked example 10: Calculating the mid-point

Calculate the coordinates of the mid-point of the line between point and point .

Draw a sketch

From the sketch, we can estimate that will lie on the -axis, with a negative -coordinate.

Assign values to and

Write down the mid-point formula

Substitute values into the mid-point formula

Write the answer

The mid-point is at .

Looking at the sketch we see that this is what we expect for the coordinates of .

For worked example 10 (calculating the mid-point) learners can check their answer using the distance formula.

Using the distance formula, we can confirm that the distances from the mid-point to each end point are equal:

and

As expected, , therefore is the mid-point.

Worked example 11: Calculating the mid-point

Find the mid-point of line , given and .

Draw a sketch

From the sketch, we can estimate that will lie in quadrant I, with positive - and -coordinates.

Assign values to and

Let the mid-point be

Write down the mid-point formula

Substitute values and simplify

Write the final answer

is the mid-point of line .

We expected to have a positive - and -coordinate and this is indeed what we have found by calculation.

Worked example 12: Using the mid-point formula

The line joining and has the mid-point . Find point .

Draw a sketch

From the sketch, we can estimate that will lie in Quadrant IV, with a positive - and negative -coordinate.

Assign values to and

Let the coordinates of be and the coordinates of be .

Write down the mid-point formula

Substitute values and solve for and

Write the final answer

The coordinates of point are .

Worked example 13: Using the mid-point formula

Points , , and are points on the Cartesian plane. Find if is a parallelogram.

Draw a sketch

Method: the diagonals of a parallelogram bisect each other, therefore the mid-point of will be the same as the mid-point of . We must first find the mid-point of . We can then use it to determine the coordinates of point .

Assign values to and

Let the mid-point of be

Write down the mid-point formula

Substitute values calculate the coordinates of

Use the coordinates of to determine

is also the mid-point of so we use and to solve for .

Substitute values and solve for and

Write the final answer

The coordinates of are .

Exercise 8.5

You are given the following diagram:

Calculate the coordinates of the mid-point () between point and point .

Let the coordinates of be and the coordinates of be .

Substitute values into the mid-point formula:

The mid-point is at .

You are given the following diagram:

Calculate the coordinates of the mid-point () between point and point .

Let the coordinates of be and the coordinates of be .

Substitute values into the mid-point formula:

The mid-point is at .

Find the mid-points of the following lines:

,

,

,

The mid-point of is . Find if is .

The mid-point formula is:

Substituting values and solving for and gives:

The coordinates of point are .

is a parallelogram with the points , and . Find point .

Draw a sketch:

The diagonals of a parallelogram bisect each other, therefore the mid-point of will be the same as the mid-point of . We must first find the mid-point of . We can then use it to determine the coordinates of point .

Use mid-point to find the coordinates of :

Solve for :

Solve for :

Therefore .

8.5 Chapter summary

Presentation: 2GDQ

A point is an ordered pair of numbers written as .

Distance is a measure of the length between two points.

The formula for finding the distance between any two points is:

The gradient between two points is determined by the ratio of vertical change to horizontal change.

The formula for finding the gradient of a line is:

A straight line is a set of points with a constant gradient between any two of the points.

The standard form of the straight line equation is .

The equation of a straight line can also be written as

If two lines are parallel, their gradients are equal.

If two lines are perpendicular, the product of their gradients is equal to .

For horizontal lines the gradient is equal to .

For vertical lines the gradient is undefined.

The formula for finding the mid-point between two points is:

Exercise 8.6

You are given the following diagram, with various points shown:

Find the coordinates of points , , , and .

From the graph we can read off the and values for each point.

, , , and

You are given the following diagram, with various points shown:

Which point lies at the coordinates ?

For this question, we are must find point .

Therefore, on the graph we can trace the and values.

Therefore, point lies at the coordinates:

You are given the following diagram, with 4 shapes drawn.

All the shapes are identical, but each shape uses a different naming convention:

Which shape uses the correct naming convention?

We recall that the correct naming convention for a shape is in alphabetical order, either clockwise or anti-clockwise around the shape.

From the diagram, we can see that only Shape Z sticks to this naming convention.

Represent the following figures in the Cartesian plane:

Triangle with , and .

Quadrilateral with , , and .

Quadrilateral with , , and .

Quadrilateral with , , and .

You are given the following diagram:

Calculate the length of line , correct to 2 decimal places.

The equation for distance is . We substitute in and and solve:

The following picture shows two points on the Cartesian plane, and .

The distance between the points is . Calculate the missing coordinate of point .

The equation for distance is . We substitute in and :

Now we re-arrange, and solve for the value of :

We now have a choice between 2 values for . From the diagram we can see that the appropriate value for this question is .

You are given the following diagram:

Calculate the gradient () of line . The coordinates are and respectively.

Therefore the gradient, , of the line is .

You are given the following diagram:

You are also told that line has a gradient, , of .

Calculate the missing co-ordinate of point .

You are given the following diagram:

You are also told that line has a gradient, , of .

Calculate the missing co-ordinate of point .

In the diagram, is the point and is the point .

Find the equation of line .

We first find the gradient of the line:

Next we note that point lies on the -axis and so is the -intercept. Therefore .

Therefore the equation of the line is: .

Calculate the length of .

You are given the following diagram:

You are also told that line runs parallel to the following line: . Point is at .

Find the equation of the line .

Let be line .

Since line is parallel to line , .

Now we can substitute in the known point () and find the -intercept of the line:

Therefore, the equation of the line is:

You are given the following diagram:

You are also told that line runs parallel to the following line: .

Calculate the missing co-ordinate of point .

Let be line .

Since line is parallel to line , .

Now we can substitute in the known point () and find the -intercept of the line:

Therefore, the equation of the line is: .

Finally, we substitute the known value for point into the equation for line :

Therefore point is at .

You are given the following diagram:

You are also told that line runs perpendicular to the following line: .

Calculate the missing co-ordinate of point .

The general form of a straight line is: .

Let line be .

Line is perpendicular to line and so .

Now we can substitute point into the equation to find the -intercept:

Next we can substitute in point to find the missing coordinate:

Therefore the missing coordinate is .

The graph here shows line, . The blue dashed line is perpendicular to .

The equation of the blue dashed line is . Point is at .

Determine the equation of line .

Let line be the blue dashed line.

Line is perpendicular to line and so .

Now we can substitute point into the equation to find the -intercept:

Therefore the equation of line is: .

You are given the following diagram:

You are also told that line has the following equation: .

Calculate the missing co-ordinate of point .

We can substitute the known value for point into the given equation for line :

You are given the following diagram:

Calculate the coordinates of the mid-point () between point and point correct to 1 decimal place.

and are points in the Cartesian plane. is the mid-point of . Find the values of and .

Determine the equations of the following straight lines:

passing through and .

parallel to and passing through .

passing through and the mid-point of where and .

In the diagram below, the vertices of the quadrilateral are , , and .

Calculate the lengths of the sides of .

To calculate the lengths of the sides we need to use the distance formula. The four sides are , , and

Are the opposite sides of parallel?

We want to know if and . We can calculate the gradient of each of the sides and then compare the gradients.

We note that and therefore the opposite sides are not parallel.

Do the diagonals of bisect each other?

To determine if the diagonals bisect each other we need to find the mid-point of and .

Therefore and the diagonals do not bisect each other.

Can you state what type of quadrilateral is? Give reasons for your answer.

This is an ordinary quadrilateral. The opposite sides are not parallel, the diagonals do not bisect each other and none of the sides are equal in length.

Consider a quadrilateral with vertices , , and .

Draw the quadrilateral.

Find the lengths of the sides of the quadrilateral.

To calculate the lengths of the sides we need to use the distance formula. The four sides are , , and

is a quadrilateral with vertices , , and .

Show by calculation that:

First draw a sketch of the quadrilateral:

To show that we need to use the distance formula to find the length of and .

Therefore sides and are equal.

To show that we need to show that .

The gradients are equal, therefore .

What type of quadrilateral is ?

An isosceles trapezium; one pair of opposite sides equal in length and one pair of opposite sides parallel.

Show that the diagonals and do not bisect each other.

To show this we need to find the mid-points of and .

, therefore the diagonals do not bisect each other.

, , and are the points , , and respectively.

Show that:

First draw a sketch:

We can use the distance formula to show that .

Since the gradients are equal .

Calculate:

We need to calculate the length of :

We need to calculate the length of :

What kind of quadrilateral is ? Give reasons for your answer.

Trapezium. One pair of opposite sides parallel.

is a parallelogram with vertices , and . Find the coordinates of by using the fact that the diagonals of a parallelogram bisect each other.

Since the diagonal bisect each other the mid-point of is equal to the mid-point of . We can first calculate the mid-point of since we have the co-ordinates of both and . We can then use that mid-point to help us find the co-ordinates of .

Solve for :

Solve for :

Therefore .

is a quadrilateral with points , , and in the Cartesian plane.

Find the length of .

First draw a sketch of the quadrilateral:

Find the gradient of .

Find the mid-point of .

Is a parallelogram? Give reasons for your answer.

We need to calculate the gradients of each of the sides to see if opposite sides are parallel. We have calculated the gradient of so we only need to check the gradients of the other three sides. However, looking at our sketch is not parallel to (you can check this by calculating the gradient of ).

And,

Therefore is not a parallelogram. Opposite sides are not parallel.

Consider triangle with vertices , and .

Sketch triangle on the Cartesian plane.

Show that is an isosceles triangle.

We need to show that two sides are equal in length. We therefore calculate the length of each of the sides of the triangle.

Two sides of the triangle are equal in length, therefore is isosceles.

Determine the coordinates of , the mid-point of .

Determine the gradient of .

Show that lies on the line that goes through and .

We have just calculated the gradient of : . We need to calculate the gradient of and :

Therefore , and are collinear.

Therefore lies on line .

has vertices , and . Show through calculation that is a right angled isosceles triangle.

First draw a sketch:

Next calculate the gradient of each of the three sides of the triangle:

Now we can check , and . As soon as we find one of these values is equal to then we have proved that the triangle is right-angled.

Therefore is right-angled, . The right-angle is .

Finally we calculate the lengths of sides and to show that the triangle is isosceles. We do not need to calculate as this is the hypotenuse of the triangle and must be longer than and .

Therefore and therefore is a right-angled, isosceles triangle.

has vertices , and . is the mid-point of and is the mid-point of . Use to prove the mid-point theorem using analytical geometry methods.

First draw a sketch:

The mid-point theorem states that the line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side. Therefore we need to show that and that .

We need to calculate the co-ordinates of mid-points and :

Now we can show that :

Finally we can use the distance formula to show that :

List two properties of a parallelogram.

Any two of the following:

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

The points , , and are the vertices of a quadrilateral. Show that the quadrilateral is a parallelogram.

We need to show that both pairs of opposite sides are parallel. So we need to calculate the gradient of each of the sides:

Therefore is a parallelogram (2 pairs opp. sides ).

The diagram shows a quadrilateral. The points and have the coordinates and respectively. The diagonals of bisect each other at right angles. is the point of intersection of line with the -axis.

Determine the gradient of .

Since the diagonals bisect each other we know that . Therefore:

Show that the equation of is given by .

From above we have that .

Substitute point into the equation:

Determine the coordinates of .

We first calculate the coordinates of :

Now we can use the coordinates of to find the coordinates of . Since is the mid-point of we can use the mid-point formula to find :

Therefore .

, and are the vertices of .

Find the length of , correct to 1 decimal place.

Calculate the gradient of .

If has coordinates , show that , and are collinear.

From the previous question we have that , therefore .

Therefore , and are collinear.

Determine the equation of line .

Substitute :

The equation of line is .

Show that is right-angled.

For a triangle to be right-angled the gradients of two of the sides must be perpendicular. We need to calculate , and .

We will start with and since we have these values from previous questions: and .

We can check if these two gradients are perpendicular:

Since the product of these two gradients is , .

Therefore is right-angled.

Given the following diagram:

If is the mid-point of , find the values of and .

Therefore and .

Find the equation of the line perpendicular to , which passes through the origin.

Let the line be .

First calculate the gradient of :

Now we can calculate the gradient of :

Therefore we have . Since the line passes through the origin the -intercept is .

Therefore the equation of the line perpendicular to , which passes through the origin is: .

Find the coordinates of the mid-point of diagonal .

Hence show that is not a parallelogram.

The mid-point of is not the mid-point of . Therefore the diagonals of the quadrilateral do not bisect each other and is not a parallelogram.

If could be moved, give its new coordinates so that would be a parallelogram.

We use the mid-point of to find the new and coordinates of :

Therefore would make a parallelogram.

A triangle has vertices , and .

Calculate the gradient of .

Prove that the triangle is right-angled at .

First draw a sketch:

For the triangle to be right-angled at , . We have from the previous question.

Therefore and is right-angled at .

Determine the length of .

Determine the equation of the line from to the mid-point of .

First find the mid-point of :

Now we can calculate the gradient of the line:

Substitute :

The equation of the the line from to the mid-point of is .

Find the area of the triangle .

First find the length of :

Now we can find the area:

A quadrilateral has vertices , , and where .

What should be so that is parallel to ?

We first draw a sketch. We note that lies somewhere on the line .

For parallel lines :

Therefore there is no value of such that will be parallel to .

We can see that this must be true since is a common point on lines and so cannot be parallel to .

What should be so that ?

But , therefore

On the Cartesian plane, the three points , and are collinear.

Find the length of .

Find the gradient of .

Find the equation of .

. Substitute :

The equation of the is .

Find the value of .

Given and .

Find the mid-point of .

Find the gradient of .

Find the gradient of the line perpendicular to .

Find the equation of the perpendicular bisector of .

From the previous question we have that . Substitute :

The equation of the perpendicular bisector of is

Find the equation of the line parallel to , passing through .

Since the line is parallel to the gradient is the same and so we have: . Substitute: :

The equation of the line parallel to and passing through is .

, , and are the vertices of parallelogram .

Determine the coordinates of .

Since is a parallelogram we know that . Therefore:

Therefore .

We can now sketch the parallelogram:

Show that is perpendicular to and state what type of quadrilateral is, other than a parallelogram.

is a rhombus since the diagonals intersect at right-angles.

Show that is a square.

We can calculate the length of each side and show that all four lengths are the same:

Therefore is a square, all the sides are equal in length.

, and are the vertices of . and are the mid-points of and respectively.

Find the gradient of .

Show that the coordinates of , the mid-point of are .

Find the length of .

Find the gradient of . Make a conjecture regarding lines and .

A conjecture regarding lines and is .

Determine the equation of .

From earlier we have the gradient of , so the equation of is: . Substitute :

The equation of is .

In the diagram points , , and are the vertices of a parallelogram.

Calculate the length of .

Find the coordinates of where the diagonals meet.

is the mid-point of both and ( is a parallelogram). Since we do not know the coordinates of we will use to find .

Find , the mid-point of .

Show that .

Calculate the coordinates of .

We can use the coordinates of to find the coordinates of :

Solve for :

Solve for :

Therefore .

The coordinates of are as follows: , and .

Through calculation, determine whether the triangle is equilateral, isosceles or scalene. Be sure to show all your working.

is an isosceles triangle.

Find the coordinates of points and , the mid-points of and .

Determine the gradient of the line .

Prove that .

The following diagram shows with . The equation of is and the equation of is . .

Write down the coordinates of .

lies on the -axis so we have . Using the equation of we get:

Prove that .

Therefore .

Write down the gradient of .

The equation of is . In standard form this is .

Therefore .

If the -coordinate of is 4, calculate the -coordinate.

We use the equation of and substitute in :

Find the equation of the line from to (the mid-point of ).

We first calculate :

Now we can find the equation of the line :

The equation of the line from to is .

The points , and are collinear.

Determine the value of .

Since , and are collinear, . Therefore:

Determine the values of and if the equation of the line passing through , and is .

Now we can rearrange the equation:

We have , therefore .

We have , therefore .

Given: .

Calculate the distance and the distance . Leave your answers in surd form.

Therefore .

Determine the coordinates of , the mid-point of .

Prove that .

Therefore .

Prove that , and are collinear.

From earlier we know that .

and is a common point.

Therefore , and are collinear.

What type of quadrilateral is ?

First draw a sketch:

From earlier we found that . Therefore the diagonals are equal in length.

Since the diagonals are equal in length we know that the quadrilateral must be a square.

We can confirm this by showing that all four sides are equal in length and that , , and .

is the mid-point of with point . Determine:

the coordinates of .

the gradient of .

the equation of the line .

the perpendicular bisector of .

The perpendicular bisector of passes through and has a gradient .

ABCD is a quadrilateral with .

Determine the coordinates of , the mid-point of .

Prove that is a parallelogram.

is a parallelogram (Diagonals bisect at ).

Find the equation of diagonal .

Determine the equation of the perpendicular bisector of .

The perpendicular bisector of passes through and has a gradient

Determine the gradient of .

Is a rhombus? Explain why or why not.

No, is not a rhombus because the diagonals do not intersect at right angles.

Find the length of .

In the diagram below, is sketched with on .

Determine the value of a in .

We can substitute into to find :

A line, , passing through , is perpendicular to . lies on . Determine the value of .

We know that , therefore .

Therefore . Substitute in to solve for :

If and are collinear, determine the value of .

, and are collinear and the equation for the line is . We can substitute into the equation for the line to solve for :

In the diagram below, and are the mid-points of and respectively.

Calculate the gradient of .

Find the equation of the line through and in the form .

Show that .

First find the coordinates of :

Next find the coordinates of :

Now we can calculate the gradient of and compare it to the gradient of :

Write down the value of the ratio: .

We note the following:

and (Mid-points)

(Mid-point theorem)

From this we can see that is twice the size of (each side of is twice the size of the same side in ). Therefore the area of is twice the area of :

Write down the coordinates of such that is a parallelogram.

We can use the fact that the diagonals of a parallelogram bisect each other to find . The mid-point of must be the same as the mid-point of .

and are points on the Cartesian plane. Determine the value of:

if , and are collinear.

First find the gradient of :

Therefore we have . Now substitute in :

Now we can substitute in to solve for :

and if is the mid-point of and .

Chapter 9: Finance and growth

9.1 Introduction

Content covered in this chapter includes the simple and compound interest formulae. These formulae are then applied to hire purchase, inflation and population growth. A short introduction to exchange rates is included.

For compound interest learners are not expected to solve for .

Discuss terminology relating to simple and compound interest such as principle amount, accumulated amount, etc.

It is very important to emphasise not rounding off calculations until final answer as this affects accuracy.

Learners should do calculations in one step using the memory function on their calculators.

In this chapter, we apply mathematical skills to everyday financial situations.

If you had , you could either keep it in your piggy bank, or deposit it into a bank account. If you deposit the money into a bank account, you are effectively lending money to the bank. Because you are lending the bank money, you can expect some extra money back. This is known as interest. Similarly, if you borrow money from a bank, then you can expect to pay interest on the loan. Interest is charged at a percentage of the money owed over the period of time it takes to pay back the loan. This means that the longer the loan exists, the more interest will have to be paid on it.

Figure 9.1: The entrance to the Johannesburg Stock Exchange (JSE) located in Sandton, Johannesburg - the financial centre of South Africa. The JSE is Africa's largest stock exchange and the largest in the world. Each month, more than 60 billion rand worth of shares are traded on the JSE.

The concept is simple, yet it is core to the world of finance. Accountants, actuaries and bankers can spend their entire working career dealing with the effects of interest on financial matters.

Interest

In finance, interest is the money charged for borrowing money. It is usually expressed as a percentage of the borrowed amount.

Some of the videos for this chapter use dollars instead of rands in the examples. Because both rands and dollars are decimal currencies you can simply change the currency symbol and the calculations will work out the same.

9.2 Simple interest

Simple interest

Simple interest is interest calculated only on the initial amount that you invested.

As an easy example of simple interest, consider how much we will get by investing for year with a bank that pays % p.a. simple interest.

At the end of the year we have

With an opening balance of at the start of the year, the closing balance at the end of the year will therefore be

The opening balance in financial calculations is often called the principal, denoted as ( in the example). The interest rate is usually labelled (% p.a. in the example and “p.a.” means per annum or per year). The interest amount is labelled ( in the example).

So we can see that

and

The above calculations give a good idea of what the simple interest formula looks like. However, the example shows an investment that lasts for only one year. If the investment or loan is over a longer period, we need to take this into account. We use the symbol to indicate time period, which must be given in years.

The general formula for calculating simple interest is

Worked example 1: Calculating interest on a deposit

Carine deposits into a special bank account which pays a simple interest rate of % p.a. for years. How much will be in her account at the end of the investment term?

Write down known values

Write down the formula

Substitute the values

Write the final answer

At the end of years, Carine will have in her bank account.

Worked example 2: Calculating interest on a loan

Sarah borrows from her neighbour at an agreed simple interest rate of % p.a. She will pay back the loan in one lump sum at the end of years. How much will she have to pay her neighbour?

Write down the known variables

Write down the formula

Substitute the values

Write the final answer

At the end of years, Sarah will pay her neighbour .

We can use the simple interest formula to find pieces of missing information. For example, if we have an amount of money that we want to invest for a set amount of time to achieve a goal amount, we can rearrange the variables to solve for the required interest rate. The same principles apply to finding the length of time we would need to invest the money, if we knew the principal and accumulated amounts and the interest rate.

Important: to get a more accurate answer, try to do all your calculations on the calculator in one go. This will prevent rounding off errors from influencing your final answer.

Worked example 3: Determining the investment period to achieve a goal amount

Prashant deposits into a bank account that pays a simple interest rate of % p.a.. How many years must he invest for to generate ?

Write down the known variables

Write down the formula

Substitute the values and solve for

Write the final answer

It will take years and months to make from at a simple interest rate of % p.a.

Worked example 4: Calculating the simple interest rate to achieve the desired growth

At what simple interest rate should Fritha invest if she wants to grow to in years?

Write down the known variables

Write down the formula

Substitute the values and solve for

Write the final answer

A simple interest rate of % p.a. will be needed when investing for years to become .

Exercise 9.1

An amount of is invested in a savings account which pays simple interest at a rate of % per annum. Calculate the balance accumulated by the end of years.

An amount of is invested in a savings account which pays simple interest at a rate of % per annum. Calculate the balance accumulated by the end of years.

Read the question carefully and write down the given information:

An amount of is invested in a savings account which pays simple interest at a rate of % per annum. Calculate the balance accumulated by the end of years.

Read the question carefully and write down the given information:

Simple interest formula:

An amount of is invested in a savings account which pays simple interest at a rate of % per annum. Calculate the balance accumulated by the end of years.

Read the question carefully and write down the given information:

Simple interest formula:

Calculate the accumulated amount in the following situations:

A loan of at a rate of % for year.

An investment of at a rate of % p.a. for years.

A bank offers a savings account which pays simple interest at a rate of % per annum. If you want to accumulate in years, how much should you invest now?

Read the question carefully and write down the given information:

Simple interest formula:

Sally wanted to calculate the number of years she needed to invest for in order to accumulate . She has been offered a simple interest rate of % p.a. How many years will it take for the money to grow to ?

It would take years for to become at % p.a.

Joseph deposited into a savings account on his son's fifth birthday. When his son turned , the balance in the account had grown to . If simple interest was used, calculate the rate at which the money was invested.

The interest rate at which the money was invested was %.

When his son was years old, Methuli made a deposit of in the bank. The investment grew at a simple interest rate and when Methuli's son was years old, the value of the investment was .

At what rate was the money invested? Give the answer correct to one decimal place.

Read the question carefully and write down the given information:

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make the subject of the formula:

When his son was years old, Phillip made a deposit of in the bank. The investment grew at a simple interest rate and when Phillip's son was years old, the value of the investment was .

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make the subject of the formula:

When his son was years old, Lefu made a deposit of in the bank. The investment grew at a simple interest rate and when Lefu's son was years old, the value of the investment was .

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make the subject of the formula:

Abdoul wants to invest at a simple interest rate of % p.a.

How many years will it take for the money to grow to ? Round up your answer to the nearest year.

Read the question carefully and write down the given information:

To calculate the number of years, we need to make the subject of the formula:

Andrew wants to invest at a simple interest rate of % p.a.

How many years will it take for the money to grow to ? Round up your answer to the nearest year.

Read the question carefully and write down the given information:

To calculate the number of years, we need to make the subject of the formula: Rounding up to the nearest year, it will take years to reach the goal of saving .

9.3 Compound interest

Compound interest allows interest to be earned on interest. With simple interest, only the original investment earns interest, but with compound interest, the original investment and the interest earned on it, both earn interest.

Compound interest is advantageous for investing money but not for taking out a loan.

Compound interest

Compound interest is the interest earned on the principal amount and on its accumulated interest.

Consider the example of invested for years with a bank that pays % p.a. compound interest.

At the end of the first year, the accumulated amount is

The amount becomes the new principal amount for calculating the accumulated amount at the end of the second year.

Similarly, we use the amount as the new principal amount for calculating the accumulated amount at the end of the third year.

Do you see a pattern?

Using the formula for simple interest, we can develop a similar formula for compound interest.

With an opening balance and an interest rate of , the closing balanced at the end of the first year is:

This is the same as simple interest because it only covers a single year. This closing balance becomes the opening balance for the second year of investment.

And similarly, for the third year

We see that the power of the term is the same as the number of years. Therefore the general formula for calculating compound interest is:

Worked example 5: Compound interest

Mpho wants to invest into an account that offers a compound interest rate of % p.a. How much money will be in the account at the end of years?

Write down the known variables

Write down the formula

Substitute the values

Write the final answer

Mpho will have in the account at the end of years.

Worked example 6: Calculating the compound interest rate to achieve the desired growth

Charlie has been given for his sixteenth birthday. Rather than spending it, he has decided to invest it so that he can put down a deposit of on a car on his eighteenth birthday. What compound interest rate does he need to achieve this growth? Comment on your answer.

Write down the known variables

Write down the formula

Substitute the values and solve for

Write the final answer and comment

Charlie needs to find an account that offers a compound interest rate of % p.a. to achieve the desired growth. A typical savings account gives a return of approximately % p.a. and an aggressive investment portfolio gives a return of approximately % p.a. It therefore seems unlikely that Charlie will be able to invest his money at an interest rate of % p.a.

The power of compound interest

To illustrate how important “interest on interest” is, we compare the difference in closing balances for an investment earning simple interest and an investment earning compound interest. Consider an amount of invested for years, at an interest rate of % p.a.

The closing balance for the investment earning simple interest is

The closing balance for the investment earning compound interest is

We plot the growth of the two investments on the same set of axes and note the significant different in their rate of change: simple interest is a straight line graph and compound interest is an exponential graph.

It is easier to see the vast difference in growth if we extend the time period to years:

Keep in mind that this is good news and bad news. When earning interest on money invested, compound interest helps that amount to grow exponentially. But if money is borrowed the accumulated amount of money owed will increase exponentially too.

This video explains the difference between simple and compound interest. Note that the video uses dollars but the calculation is the same for rands.

Video: 2GH3

Exercise 9.2

An amount of is invested in a savings account which pays a compound interest rate of % p.a. Calculate the balance accumulated by the end of years.

An amount of is invested in a savings account which pays a compound interest rate of % p.a.

Calculate the balance accumulated by the end of years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information:

The accumulated amount is:

An amount of is invested in a savings account which pays a compound interest rate of % p.a.

Calculate the balance accumulated by the end of years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information:

The accumulated amount is:

Nicola wants to invest some money at a compound interest rate of % p.a. How much money (to the nearest rand) should be invested if she wants to reach a sum of in five years time?

Thobeka wants to invest some money at a compound interest rate of % p.a.

How much money should be invested if she wants to reach a sum of in years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information:

To determine the amount she must invest, we need to make the subject of the formula:

She must invest .

Likengkeng wants to invest some money at a compound interest rate of % p.a.

How much money should be invested if she wants to reach a sum of in years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information:

To determine the amount she must invest, we need to make the subject of the formula:

Morgan invests into an account which pays out a lump sum at the end of years. If he gets at the end of the period, what compound interest rate did the bank offer him?

The interest rate is % p.a

Kabir invests into an account which pays out a lump sum at the end of years.

If he gets at the end of the period, what compound interest rate did the bank offer him? Give the answer correct to one decimal place.

Read the question carefully and write down the given information:

To calculate the interest rate, we need to make the subject of the formula:

Bongani invests into an account which pays out a lump sum at the end of years.

If he gets at the end of the period, what compound interest rate did the bank offer him? Give the answer correct to one decimal place.

Read the question carefully and write down the given information:

To calculate the interest rate, we need to make the subject of the formula:

9.4 Calculations using simple and compound interest

Hire purchase

As a general rule, it is not wise to buy items on credit. When buying on credit you have to borrow money to pay for the object, meaning you will have to pay more for it due to the interest on the loan. That being said, occasionally there are appliances, such as a fridge, that are very difficult to live without. Most people don't have the cash up front to purchase such items, so they buy it on a hire purchase agreement.

A hire purchase agreement is a financial agreement between the shop and the customer about how the customer will pay for the desired product. The interest on a hire purchase loan is always charged at a simple interest rate and only charged on the amount owing. Most agreements require that a deposit is paid before the product can be taken by the customer. The principal amount of the loan is therefore the cash price minus the deposit. The accumulated loan will be worked out using the number of years the loan is needed for. The total loan amount is then divided into monthly payments over the period of the loan.

Hire purchase is charged at a simple interest rate. When you are asked a hire purchase question, don't forget to always use the simple interest formula.

This video explains hire purchase and shows some examples of hire purchase calculations.

Video: 2GHF

Worked example 7: Hire purchase

Troy wants to buy an additional screen for his computer which he saw advertised for on the internet. There is an option of paying a % deposit and then making monthly payments using a hire purchase agreement, where interest is calculated at % p.a. simple interest. Calculate what Troy's monthly payments will be.

Write down the known variables

A new opening balance is required, as the % deposit is paid in cash.

Write down the formula

Substitute the values

Calculate the monthly repayments on the hire purchase agreement

Write the final answer

Troy's monthly payment is .

A shop can also add a monthly insurance premium to the monthly instalments. This insurance premium will be an amount of money paid monthly and gives the customer more time between a missed payment and possible repossession of the product.

The monthly payment is also called the monthly instalment.

Worked example 8: Hire purchase with extra conditions

Cassidy wants to buy a TV and decides to buy one on a hire purchase agreement. The TV's cash price is . She will pay it off over months at an interest rate of % p.a. An insurance premium of is added to every monthly payment. How much are her monthly payments?

Write down the known variables

The question does not mention a deposit, therefore we assume that Cassidy did not pay one.

Write down the formula

Substitute the values

Calculate the monthly repayments on the hire purchase agreement

Add the insurance premium

Write the final answer

Cassidy will pay per month for months until her TV is paid off.

Exercise 9.3

Angelique wants to buy a microwave on a hire purchase agreement. The cash price of the microwave is . She is required to pay a deposit of % and pay the remaining loan amount off over months at an interest rate of % p.a.

What is the principal loan amount?

First calculate the amount for the deposit:

To determine the principal loan amount, we must subtract the deposit amount from the cash price:

What is the accumulated loan amount?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

What are Angelique's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount by the total number of months:

What is the total amount she has paid for the microwave?

To determine the total amount paid, we add the accumulated loan amount and the deposit:

Nyakallo wants to buy a television on a hire purchase agreement. The cash price of the television is . She is required to pay a deposit of % and pay the remaining loan amount off over months at an interest rate of % p.a.

What is the principal loan amount?

First calculate the amount for the deposit:

To determine the principal loan amount, we must subtract the deposit amount from the cash price:

What is the accumulated loan amount?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

What are Nyakallo's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount by the total number of months:

What is the total amount she has paid for the television?

To determine the total amount paid we add the accumulated loan amount and the deposit:

A company wants to purchase a printer. The cash price of the printer is . A deposit of % is required on the printer. The remaining loan amount will be paid off over months at an interest rate of % p.a.

What is the principal loan amount?

To calculate the principal loan amount, we first calculate the amount for the deposit and then subtract the deposit amount from the cash price:

What is the accumulated loan amount?

Remember that hire purchase uses simple interest. We write down the given information and then substitute these values into the simple interest formula.

How much will the company pay each month?

To determine the monthly payment amount (how much the company pays each month), we divide the accumulated amount by the total number of months:

What is the total amount the company paid for the printer?

To determine the total amount paid we add the accumulated loan amount and the deposit:

Sandile buys a dining room table costing on a hire purchase agreement. He is charged an interest rate of % p.a. over years.

How much will Sandile pay in total?

The question does not mention a deposit so we assume Sandile did not pay one. We write down the given information and then use the simple interest formula to calculate the accumulated amount.

How much interest does he pay?

To calculate the total interest paid we subtract the cash price from the accumulated amount.

What is his monthly instalment?

To determine the monthly payment amount, we divide the accumulated amount by the total number of months:

Mike buys a table costing on a hire purchase agreement. He is charged an interest rate of % p.a. over years.

How much will Mike pay in total?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount:

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount by the total number of months:

Talwar buys a cupboard costing on a hire purchase agreement. He is charged an interest rate of % p.a. over years.

How much will Talwar pay in total?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount:

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount by the total number of months:

A lounge suite is advertised for sale on TV, to be paid off over months at per month.

Assuming that no deposit is needed, how much will the buyer pay for the lounge suite once it has been paid off?

If the interest rate is % p.a., what is the cash price of the suite?

Two stores are offering a fridge and washing machine combo package. Store A offers a monthly payment of over months. Store B offers a monthly payment of over months.

If both stores offer % interest, which store should you purchase the fridge and washing machine from if you want to pay the least amount of interest?

To calculate the interest paid at each store we need to first find the cash price of the fridge and washing machine.

Store A:

Therefore the interest is

Store B:

Therefore the interest is

If you want to pay the least amount in interest you should purchase the fridge and washing machine from store A.

Tlali wants to buy a new computer and decides to buy one on a hire purchase agreement. The computers cash price is . He will pay it off over months at an interest rate of % p.a. An insurance premium of is added to every monthly payment. How much are his monthly payments?

The question does not mention a deposit, therefore we assume that Tlali did not pay one.

The monthly payment is:

Add the insurance premium:

Richard is planning to buy a new stove on hire purchase. The cash price of the stove is . He has to pay a % deposit and then pay the remaining amount off over months at an interest rate of % p.a. An insurance premium of is added to every monthly payment. Calculate Richard's monthly payments.

Calculate the accumulated amount:

Calculate the monthly repayments on the hire purchase agreement:

Add the insurance premium:

Inflation

There are many factors that influence the change in price of an item, one of them is inflation. Inflation is the average increase in the price of goods each year and is given as a percentage. Since the rate of inflation increases year on year, it is calculated using the compound interest formula.

Worked example 9: Calculating future cost based on inflation

Milk costs for two litres. How much will it cost in years time if the inflation rate is % p.a.?

Write down the known variables

Write down the formula

Substitute the values

Write the final answer

In four years time, two litres of milk will cost .

Worked example 10: Calculating past cost based on inflation

A box of chocolates costs today. How much did it cost years ago if the average rate of inflation was % p.a.?

Write down the known variables

Write down the formula

Substitute the values and solve for

Write the final answer

Three years ago, the box of chocolates would have cost .

Exercise 9.4

The price of a bag of apples is . How much will it cost in years time if the inflation rate is % p.a.?

Read the question carefully and write down the given information:

To determine the future cost, we use the compound interest formula:

The price of a bag of potatoes is .

How much will it cost in years time if the inflation rate is % p.a.?

Read the question carefully and write down the given information:

To determine the future cost, we use the compound interest formula:

The price of a box of popcorn is . How much will it cost in years time if the inflation rate is % p.a.?

Read the question carefully and write down the given information:

To determine the future cost, we use the compound interest formula:

A box of raisins costs today. How much did it cost years ago if the average rate of inflation was % p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

We use the compound interest formula and make the subject:

A box of biscuits costs today. How much did it cost years ago if the average rate of inflation was % p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

We use the compound interest formula and make the subject:

If the average rate of inflation for the past few years was % p.a. and your water and electricity account is on average, what would you expect to pay in years time?

The price of popcorn and a cooldrink at the movies is now . If the average rate of inflation is % p.a. what was the price of popcorn and cooldrink years ago?

Population growth

Family trees increase exponentially as every person born has the ability to start another family. For this reason we calculate population growth using the compound interest formula.

Worked example 11: Population growth

If the current population of Johannesburg is , and the average rate of population growth in South Africa is % p.a., what can city planners expect the population of Johannesburg to be in years?

Write down the known variables

Write down the formula

Substitute the values

Write the final answer

City planners can expect Johannesburg's population to be in ten years time.

Exercise 9.5

The current population of Durban is and the average rate of population growth in South Africa is % p.a.

What can city planners expect the population of Durban to be in years time? Round your answer to the nearest integer.

Read the question carefully and write down the given information:

We use the following formula to determine the expected population for Durban:

The current population of Polokwane is and the average rate of population growth in South Africa is % p.a.

What can city planners expect the population of Polokwane to be in years time? Round your answer to the nearest integer.

Read the question carefully and write down the given information:

We use the following formula to determine the expected population for Polokwane:

A small town in Ohio, USA is experiencing a huge increase in births. If the average growth rate of the population is % p.a., how many babies will be born to the residents in the next years?

There will be roughly babies born in the next two years.

9.5 Foreign exchange rates

Different countries have their own currencies. In England, a Big Mac from McDonald's costs , in South Africa it costs and in Norway it costs  kr. The meal is the same in all three countries but in some places it costs more than in others. If and , this means that a Big Mac in England costs and a Big Mac in Norway costs .

Exchange rates affect a lot more than just the price of a Big Mac. The price of oil increases when the South African rand weakens. This is because when the rand is weaker, we can buy less of other currencies with the same amount of money.

A currency gets stronger when money is invested in the country. When we buy products that are made in South Africa, we are investing in South African business and keeping the money in the country. When we buy products imported from other countries, we are investing money in those countries and as a result, the rand will weaken. The more South African products we buy, the greater the demand for them will be and more jobs will become available for South Africans. Local is lekker!

The three currencies you are most likely to see are the British pound (£), the American dollar () and the euro (\euro).

This video explains exchange rates and shows some examples of exchange rate calculations.

Video: 2GJ6

Worked example 12: Foreign exchange rates

Saba wants to travel to see her family in Spain. She has been given spending money. How many euros can she buy if the exchange rate is currently \euro{} ?

Write down the equation

Let the equivalent amount in euros be

Write the final answer

Saba can buy \euro{} with .

Exercise 9.6

Bridget wants to buy an iPod that costs , with the exchange rate currently at . She estimates that the exchange rate will drop to in a month.

How much will the iPod cost in rands, if she buys it now?

Cost in rands = (cost in pounds) exchange rate.

How much will she save if the exchange rate drops to ?

Cost in rands

So she will save (Saving )

How much will she lose if the exchange rate moves to ?

Cost in rands

So she will lose (Loss )

Mthuli wants to buy a television that costs , with the exchange rate currently at . He estimates that the exchange rate will drop to in a month.

How much will the television cost in rands, if he buys it now?

How much will he save if the exchange rate drops to ?

Therefore the amount he will have saved is:

How much will he lose if the exchange rate moves to ?

Therefore the amount he will lose is:

Nthabiseng wants to buy an iPad that costs , with the exchange rate currently at . She estimates that the exchange rate will drop to in a month.

How much will the iPad cost, in rands, if she buys it now?

How much will she save if the exchange rate drops to ?

Therefore the amount she will have saved is:

How much will she lose if the exchange rate moves to ?

Therefore the amount she will lose is:

Study the following exchange rate table:

Country

Currency

Exchange Rate

United Kingdom (UK)

Pounds (£)

United States (USA)

Dollars ()

In South Africa the cost of a new Honda Civic is . In England the same vehicle costs and in the USA . In which country is the car the cheapest?

To answer this question we work out the cost of the car in rand for each country and then compare the three answers to see which is the cheapest. Cost in rands = cost in currency times exchange rate.

Cost in UK:

Cost in USA:

Comparing the three costs we find that the car is the cheapest in the USA.

Sollie and Arinda are waiters in a South African restaurant attracting many tourists from abroad. Sollie gets a tip from a tourist and Arinda gets . Who got the better tip?

Sollie:

Arinda: .

Therefore Sollie got the better tip. He got cents more than Arinda.

Yaseen wants to buy a book online. He finds a publisher in London selling the book for . This publisher is offering free shipping on the product.

He then finds the same book from a publisher in New York for with a shipping fee of .

Next he looks up the exchange rates to see which publisher has the better deal. If and , which publisher should he buy the book from?

London publisher:

New York publisher: .

Therefore Yaseen should buy the book from the New York publisher.

Mathe is saving up to go visit her friend in Germany. She estimates the total cost of her trip to be . The exchange rate is currently .

Her friend decides to help Mathe out by giving her . How much (in rand) does Mathe now need to save up?

We first calculate how much Mathe's friend will give her in rands:

.

Therefore Mathe now needs to save up: .

Lulamile and Jacob give tours over the weekends. They do not charge for these tours but instead accept tips from the group. The table below shows the total amount of tips they receive from various tour groups.

Group | Total tips
British tourists |
Japanese tourists |
American tourists |
Dutch tourists |
Brazilian tourists |
Australian tourists |
South African tourists |

The current exchange rates are:

Which group of tourists tipped the most? How much did they tip (give your answer in rand)?

We need to calculate the value of each tip in rand:

Group | Total tips | Value of tip in rands
British tourists |  |
Japanese tourists |  |
American tourists |  |
Dutch tourists |  |
Brazilian tourists |  |
Australian tourists |  |
South African tourists |  |

The Brazilian tourists tipped the most. The rand value of their tip was .

Which group of tourists tipped the least? How much did they tip (give your answer in rand)?

The Japanese tourists tipped the least. The rand value of their tip was .

Kayla is planning a trip to visit her family in Malawi followed by spending some time in Tanzania at the Serengeti. She will first need to convert her South African rands into the Malawian kwacha. After that she will convert her remaining Malawian kwacha into Tanzanian shilling.

She looks up the current exchange rates and finds the following information:

She starts off with in South Africa. In Malawi she spends . When she converts the remaining Malawian kwacha to Tanzanian shilling, how much money does she have (in Tanzanian shilling)?

We first convert from rands to Malawian kwacha:

She spends of this and so she has to convert to Tanzanian shillings.

Now we can convert from Malawian kwacha to Tanzanian shillings:

So she will have to spend in Tanzania.

9.6 Chapter summary

Presentation: 2GJH

There are two types of interest rates:

Simple interest

Compound interest

Where:

Hire purchase loan repayments are calculated using the simple interest formula on the cash price minus the deposit. Monthly repayments are calculated by dividing the accumulated amount by the number of months for the repayment.

Population growth and inflation are calculated using the compound interest formula.

Foreign exchange rate is the price of one currency in terms of another.

Exercise 9.7

An amount of is invested in a savings account which pays simple interest at a rate of % p.a.. Calculate the balance accumulated by the end of years.

Read the question carefully and write down the given information:

Simple interest formula:

An amount of is invested in a savings account which pays simple interest at a rate of % p.a.. Calculate the balance accumulated by the end of years.

Read the question carefully and write down the given information:

Simple interest formula:

Adam opens a savings account when he is . He would like to have by the time he is . If the savings account offers simple interest at a rate of % per annum, how much money should he invest now to reach his goal?

Read the question carefully and write down the given information:

Simple interest formula:

When his son was years old, Dumile made a deposit of in the bank. The investment grew at a simple interest rate and when Dumile's son was years old, the value of the investment was .

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make the subject of the formula:

When his son was years old, Jared made a deposit of in the bank. The investment grew at a simple interest rate and when Jared's son was years old, the value of the investment was .

At what rate was the money invested? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

The question says that the investment “grew at a simple interest rate”, so we must use the simple interest formula. To calculate the interest rate, we need to make the subject of the formula:

Sehlolo wants to invest at a simple interest rate of % p.a.

How many years will it take for the money to grow to ? Round up your answer to the nearest year.

Read the question carefully and write down the given information:

To calculate the number of years, we need to make the subject of the formula: Rounding up to the nearest year, it will take years to reach the goal of saving .

Mphikeleli wants to invest at a simple interest rate of % p.a.

How many years will it take for the money to grow to ? Round up your answer to the nearest year.

Read the question carefully and write down the given information:

To calculate the number of years, we need to make the subject of the formula: Rounding up to the nearest year, it will take years to reach the goal of saving .

An amount of is invested in an account which pays simple interest at a rate of % per annum. Calculate the amount of interest accumulated at the end of years.

Read the question carefully and write down the given information:

Therefore the interest earned is

An amount of is invested in a savings account which pays a compound interest rate of % p.a.

Calculate the balance accumulated by the end of years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information:

The accumulated amount is:

An amount of is invested in a savings account which pays a compound interest rate of % p.a.

Calculate the balance accumulated by the end of years. As usual with financial calculations, round your answer to two decimal places, but do not round off until you have reached the solution.

Read the question carefully and write down the given information:

The accumulated amount is:

Emma wants to invest some money at a compound interest rate of % p.a.

How much money should be invested if she wants to reach a sum of in years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information:

To determine the amount she must invest, we need to make the subject of the formula:

Limpho wants to invest some money at a compound interest rate of % p.a.

How much money should be invested if she wants to reach a sum of in years' time? Round up your answer to the nearest rand.

Read the question carefully and write down the given information:

To determine the amount she must invest, we need to make the subject of the formula:

Calculate the compound interest for the following problems.

A loan for years at % p.a.

A investment for years at % p.a.

A loan for year at % p.a.

Ali invests into an account which pays out a lump sum at the end of years.

If he gets at the end of the period, what compound interest rate did the bank offer him? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

Christopher invests into an account which pays out a lump sum at the end of years.

If he gets at the end of the period, what compound interest rate did the bank offer him? Give your answer correct to one decimal place.

Read the question carefully and write down the given information:

Calculate how much you will earn if you invested for year at the following interest rates:

% simple interest

% compound interest

Bianca has to invest for years. Bank A offers a savings account which pays simple interest at a rate of % per annum, whereas Bank B offers a savings account paying compound interest at a rate of % per annum. Which account would leave Bianca with the highest accumulated balance at the end of the year period?

Bank A:

Bank B:

She should choose Bank B as it will give her more money after 3 years.

Given:

A loan of for a year at an interest rate of % p.a.

How much simple interest is payable on the loan?

How much compound interest is payable on the loan?

is invested at an interest rate of % per annum.

Complete the following table.

Number of years
Simple interest
Compound interest

We need to calculate the amount accumulated if the interest rate is simple interest. We use to do this.

We also need to calculate the amount accumulated if the interest rate is compound interest. We use to do this.

For both cases we note that:

Number of years
Simple interest
Compound interest

Discuss:

Which type of interest would you like to use if you are the borrower?

Simple interest. Interest is only calculated on the principal amount and not on the interest earned during prior periods. This will lead to the borrower paying less interest.

Which type of interest would you like to use if you were the banker?

Compound interest. Interest is calculated from the principal amount as well as interest earned from prior periods. This will lead to the banker getting more money for the bank.

Portia wants to buy a television on a hire purchase agreement. The cash price of the television is . She is required to pay a deposit of % and pay the remaining loan amount off over months at an interest rate of % p.a.

What is the principal loan amount?

First calculate the amount for the deposit:

To determine the principal loan amount, we must subtract the deposit amount from the cash price:

What is the accumulated loan amount?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

What are Portia's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount by the total number of months:

What is the total amount she has paid for the television?

To determine the total amount paid we add the accumulated loan amount and the deposit:

Gabisile wants to buy a heater on a hire purchase agreement. The cash price of the heater is . She is required to pay a deposit of % and pay the remaining loan amount off over months at an interest rate of % p.a.

What is the principal loan amount?

First calculate the amount for the deposit:

To determine the principal loan amount, we must subtract the deposit amount from the cash price:

What is the accumulated loan amount?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

What are Gabisile's monthly repayments?

To determine the monthly payment amount, we divide the accumulated amount by the total number of months:

What is the total amount she has paid for the heater?

To determine the total amount paid we add the accumulated loan amount and the deposit:

Khayalethu buys a couch costing on a hire purchase agreement. He is charged an interest rate of % p.a. over years.

How much will Khayalethu pay in total?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount:

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount by the total number of months:

Jwayelani buys a sofa costing on a hire purchase agreement. He is charged an interest rate of % p.a. over years.

How much will Jwayelani pay in total?

Read the question carefully and write down the given information:

To determine the accumulated loan amount, we use the simple interest formula:

How much interest does he pay?

To determine the interest amount, we subtract the principal amount from the accumulated amount:

What is his monthly instalment?

To determine the monthly instalment amount, we divide the accumulated amount by the total number of months:

Bonnie bought a stove for . After years she had finished paying for it and the interest that was charged for hire purchase. Determine the rate of simple interest that was charged.

A new furniture store has just opened in town and is offering the following special:

Purchase a lounge suite, a bedroom suite and kitchen appliances (fridge, stove, washing machine) for just and receive a free microwave. No deposit required, year payment plan available. Interest charged at just % p.a.

Babelwa purchases all the items on hire purchase. She decides to pay a deposit. The store adds in an insurance premium of per month.

What is Babelwa's monthly payment on the items?

Calculate the accumulated amount:

Calculate the monthly repayments on the hire purchase agreement:

Add the insurance premium:

The price of 2 litres of milk is . How much will it cost in years time if the inflation rate is % p.a.?

Read the question carefully and write down the given information:

To determine the future cost, we use the compound interest formula:

The price of a bottle of juice is . How much will the juice cost in years time if the inflation rate is % p.a.?

Read the question carefully and write down the given information:

To determine the future cost, we use the compound interest formula:

A box of fruity-chews costs today. How much did it cost years ago if the average rate of inflation was % p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

We use the compound interest formula and make the subject:

A box of smarties costs today. How much did the same box cost years ago if the average rate of inflation was % p.a.? Round your answer to 2 decimal places.

Read the question carefully and write down the given information:

We use the compound interest formula and make the subject:

According to the latest census, South Africa currently has a population of .

If the annual growth rate is expected to be %, calculate how many South Africans there will be in years time (correct to the nearest hundred thousand).

If it is found after years that the population has actually increased by million to million, what was the growth rate?

The current population of Cape Town is and the average rate of population growth in South Africa is % p.a.

What can city planners expect the population of Cape Town to be in years time? Note: Round your answer to the nearest integer.

Read the question carefully and write down the given information:

We use the following formula to determine the expected population for Cape Town:

The current population of Pretoria is and the average rate of population growth in South Africa is % p.a.

What will the population of Pretoria be in years time? Note: Round your answer to the nearest integer.

Read the question carefully and write down the given information:

We use the following formula to determine the expected population for Pretoria:

Monique wants to buy an iPad that costs , with the exchange rate currently at = . She estimates that the exchange rate will drop to in a month.

How much will the iPad cost in rands, if she buys it now?

How much will she save if the exchange rate drops to ?

Therefore the amount she will have saved is:

How much will she lose if the exchange rate moves to ?

Therefore the amount she will lose is:

Xolile wants to buy a CD player that costs , with the exchange rate currently at . She estimates that the exchange rate will drop to in a month.

How much will the CD player cost in rands, if she buys it now?

How much will she save if the exchange rate drops to ?

Therefore the amount she will have saved is:

How much will she lose if the exchange rate moves to ?

Therefore the amount she will lose is:

Alison is going on holiday to Europe. Her hotel will cost per night. How much will she need, in rands, to cover her hotel bill, if the exchange rate is ?

Cost in rands = cost in Euros exchange rate

Jennifer is buying some books online. She finds a publisher in the UK selling the books for .

She then finds the same books from a publisher in the USA for .

Next she looks up the exchange rates to see which publisher has the better deal. If and , which publisher should she buy the books from?

UK publisher:

USA publisher: .

Therefore Jennifer should buy the books from the UK publisher.

Bonani won a trip to see Machu Picchu in Peru followed by a trip to Brazil for the carnival. He is given to spend while on the trip.

He then looks up the current exchange rates and finds the following information:

In Peru he spends . When he converts the remaining Peruvian sol to Brazilian real, how much money does he have (in Brazilian real)?

We first convert from rand to Peruvian sol:

He spends of this and so he has to convert to Brazilian real.

Now we can convert from Peruvian sol to Brazilian real:

So he will have to spend in Brazil.

If the exchange rate to the rand for the Japanese yen is ¥  = and for the Australian dollar is  AUD = , determine the exchange rate between the Australian dollar and the Japanese yen.

Khetang has just been to Europe to work for a few months. He returns to South Africa with to invest in a savings account.

His bank offers him a savings account which pays % compound interest per annum. The bank converts Khetang's Euros to rands at an exchange rate of .

If Khetang invests his money for years, how much interest does he earn on his investment?

We first convert from Euros to rands: .

Now we can calculate how much Khetang earns.

Interest earned:

Chapter 10: Statistics

This chapter covers revision of central tendency in ungrouped data and then extends this to measures of central tendency in grouped data. The range is revised and extended to include percentiles, quartiles, interquartile and semi interquartile range. The five number summary and box and whisker diagram is introduced here. Finally statistical summaries are applied to data to make meaningful comments on the context associated with the data.

Intervals for grouped data should be given using inequalities () rather than 0 - 19.

Discuss the misuse of statistics in the real world and encourage awareness.

You can find data sets and statistics relevant to South Africa from the statssa website.

When running an experiment or conducting a survey we can potentially end up with many hundreds, thousands or even millions of values in the resulting data set. Too much data can be overwhelming and we need to reduce them or represent them in a way that is easier to understand and communicate.

Statistics is about summarising data. The methods of statistics allow us to represent the essential information in a data set while disregarding the unimportant information. We have to be careful to make sure that we do not accidentally throw away some of the important aspects of a data set.

By applying statistics properly we can highlight the important aspects of data and make the data easier to interpret. By applying statistics poorly or dishonestly we can also hide important information and let people draw the wrong conclusions.

In this chapter we will look at a few numerical and graphical ways in which data sets can be represented, to make them easier to interpret.

Figure 10.1: Statistics is used by various websites to show users who is viewing their content.

10.1 Collecting data

Data

Data refers to the pieces of information that have been observed and recorded, from an experiment or a survey.

The word data is the plural of the word datum, and therefore one should say, “the data are” and not “the data is”.

We distinguish between two main types of data: quantitative and qualitative.

Quantitative data

Quantitative data are data that can be written as numbers.

Quantitative data can be discrete or continuous. Discrete quantitative data can be represented by integers and usually occur when we count things, for example, the number of learners in a class, the number of molecules in a chemical solution, or the number of SMS messages sent in one day.

Continuous quantitative data can be represented by real numbers, for example, the height or mass of a person, the distance travelled by a car, or the duration of a phone call.

Qualitative data

Qualitative data are data that cannot be written as numbers.

Two common types of qualitative data are categorical and anecdotal data. Categorical data can come from one of a limited number of possibilities, for example, your favourite cooldrink, the colour of your cell phone, or the language that you learnt to speak at home.

Anecdotal data take the form of an interview or a story, for example, when you ask someone what their personal experience was when using a product, or what they think of someone else's behaviour.

Categorical qualitative data are sometimes turned into quantitative data by counting the number of times that each category appears. For example, in a class with learners, we ask everyone what the colours of their cell phones are and get the following responses:

black

black

black

white

purple

red

red

black

black

black

white

white

black

black

black

black

purple

black

black

white

purple

black

red

red

white

black

orange

orange

black

white

This is a categorical qualitative data set since each of the responses comes from one of a small number of possible colours.

We can represent exactly the same data in a different way, by counting how many times each colour appears.

Colour

Count

black

white

red

purple

orange

This is a discrete quantitative data set since each count is an integer.

Worked example 1: Qualitative and quantitative data

Thembisile is interested in becoming an airtime reseller to his classmates. He would like to know how much business he can expect from them. He asked each of his classmates how many SMS messages they sent during the previous day. The results were:

Is this data set qualitative or quantitative? Explain your answer.

The number of SMS messages is a count represented by an integer, which means that it is quantitative and discrete.

Worked example 2: Qualitative and quantitative data

Thembisile would like to know who the most popular cellular provider is among learners in his school. This time Thembisile randomly selects learners from the entire school and asks them which cellular provider they currently use. The results were:

Cell C

Vodacom

Vodacom

MTN

Vodacom

MTN

MTN

Virgin Mobile

Cell C

8-ta

Vodacom

MTN

Vodacom

Vodacom

MTN

Vodacom

Vodacom

Vodacom

Virgin Mobile

MTN

Is this data set qualitative or quantitative? Explain your answer.

Since each response is not a number, but one of a small number of possibilities, these are categorical qualitative data.

Exercise 10.1

The following data set of dreams that learners have was collected from Grade 12 learners just after their final exams.

Categorise the data set.

This data set cannot be written as numbers and so must be qualitative.

This data set is anecdotal since it takes the form of a story.

Therefore the data set is qualitative anecdotal.

The following data set of sweets in a packet was collected from visitors to a sweet shop.

Categorise the data set.

This data set is a set of numbers and so must be quantitative.

This data set is discrete since it can be represented by integers and is a count of the number of sweets.

Therefore the data set is quantitative discrete.

The following data set of questions answered correctly was collected from a class of maths learners.

Categorise the data set.

This data set is a set of numbers and so must be quantitative.

This data set is discrete since it can be represented by integers and is a count of the number of questions answered correctly.

Therefore the data set is quantitative discrete.

10.2 Measures of central tendency

Mean

Mean

The mean is the sum of a set of values, divided by the number of values in the set. The notation for the mean of a set of values is a horizontal bar over the variable used to represent the set, for example . The formula for the mean of a data set is:

The mean is sometimes also called the average or the arithmetic mean.

Worked example 3: Calculating the mean

What is the mean of the data set ?

Calculate the sum of the data

Divide by the number of values in the data set to get the mean

Since there are values in the data set, the mean is:

Median

Median

The median of a data set is the value in the central position, when the data set has been arranged from the lowest to the highest value.

Note that exactly half of the values from the data set are less than the median and the other half are greater than the median.

To calculate the median of a quantitative data set, first sort the data from the smallest to the largest value and then find the value in the middle. If there is an odd number of values in the data set, the median will be equal to one of the values in the data set. If there is an even number of values in the data set, the median will lie halfway between two values in the data set.

Worked example 4: Median for an odd number of values

What is the median of ?

Sort the values

The values in the data set, arranged from the smallest to the largest, are

Find the number in the middle

There are values in the data set. Since there are an odd number of values, the median will be equal to the value in the middle, namely, in the fourth position. Therefore the median of the data set is .

Worked example 5: Median for an even number of values

What is the median of ?

Sort the values

The values in the data set, arranged from the smallest to the largest, are

Find the number in the middle

There are values in the data set. Since there are an even number of values, the median will be halfway between the two values in the middle, namely, between the fourth and fifth positions. The value in the fourth position is and the value in the fifth position is . The median lies halfway between these two values and is therefore

Mode

Mode

The mode of a data set is the value that occurs most often in the set. The mode can also be described as the most frequent or most common value in the data set.

To calculate the mode, we simply count the number of times that each value appears in the data set and then find the value that appears most often.

A data set can have more than one mode if there is more than one value with the highest count. For example, both and are modes in the data set . If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

The following video explains how to calculate the mean, median and mode of a data set.

Video: 2GM5

Worked example 6: Finding the mode

Find the mode of the data set .

Count the number of times that each value appears in the data set

Value

Count

Find the value that appears most often

From the table above we can see that is the only value that appears times. All the other values appear less than 3 times. Therefore the mode of the data set is .

One problem with using the mode as a measure of central tendency is that we can usually not compute the mode of a continuous data set. Since continuous values can lie anywhere on the real line, any particular value will almost never repeat. This means that the frequency of each value in the data set will be and that there will be no mode. We will look at one way of addressing this problem in the section on grouping data.

Worked example 7: Comparison of measures of central tendency

There are regulations in South Africa related to bread production to protect consumers. By law, if a loaf of bread is not labelled, it must weigh , with the leeway of percent under or percent over. Vishnu is interested in how a well-known, national retailer measures up to this standard. He visited his local branch of the supplier and recorded the masses of different loaves of bread for one week. The results, in grams, are given below:

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Is this data set qualitative or quantitative? Explain your answer.

Determine the mean, median and mode of the mass of a loaf of bread for each day of the week. Give your answer correct to 1 decimal place.

Based on the data, do you think that this supplier is providing bread within the South African regulations?

Qualitative or quantitative?

Since each mass can be represented by a number, the data set is quantitative. Furthermore, since a mass can be any real number, the data are continuous.

Calculate the mean

In each column (for each day of the week), we add up the measurements and divide by the number of measurements, .

For Monday, the sum of the measured values is and so the mean for Monday is

In the same way, we can compute the mean for each day of the week. See the table below for the results.

Calculate the median

In each column we sort the numbers from lowest to highest and find the value in the middle. Since there are an even number of measurements (), the median is halfway between the two numbers in the middle.

For Monday, the sorted list of numbers is

The two numbers in the middle are and and so the median is

In the same way, we can compute the median for each day of the week:

Day

Mean

Median

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

From the above calculations we can see that the means and medians are close to one another, but not quite equal. In the next worked example we will see that the mean and median are not always close to each other.

Determine the mode

Since the data are continuous we cannot compute the mode. In the next section we will see how we can group data in order to make it possible to compute an approximation for the mode.

Conclusion: Is the supplier reliable?

From the question, the requirements are that the mass of a loaf of bread be between minus %, which is , and plus %, which is . Since every one of the measurements made by Vishnu lies within this range and since the means and medians are all close to , we can conclude that the supplier is reliable.

Outlier

An outlier is a value in the data set that is not typical of the rest of the set. It is usually a value that is much greater or much less than all the other values in the data set.

Worked example 8: Effect of outliers on mean and median

The heights of learners are measured in centimetres to obtain the following data set:

Afterwards, we include one more learner in the group, who is exceptionally tall at .

Compare the mean and median of the heights of the learners before and after the eleventh learner was included.

Calculate the mean of the first learners

Calculate the mean of all learners

From this we see that the average height changes by when we introduce the outlier value (the tall person) to the data set.

Calculate the median of the first learners

To find the median, we need to sort the data set:

Since there are an even number of values, , the median lies halfway between the fifth and sixth values:

Calculate the median of all learners

After adding the tall learner, the sorted data set is

Now, with values, the median is the sixth value: . So, the median changes by only when we add the outlier value to the data set.

In general, the median is less affected by the addition of outliers to a data set than the mean is. This is important because it is quite common that outliers are measured during an experiment, because of problems with the equipment or unexpected interference.

Exercise 10.2

Calculate the mean of the following data set:

. Round your answer to 1 decimal place.

The mean is: .

Calculate the median of the following data set:

.

We first need to order the data set:

.

Since there are an even number of values in this data set (10) the median lies between the fifth and sixth place:

The median is: 13.

Calculate the mode of the following data set:

We first sort the data set: . The mode is the value that occurs most often in the data set.

Therefore the mode is: 6

Calculate the mean, median and mode of the following data sets:

The data set is already ordered.

Since there is an odd number of values in this data set the median lies at the fifth number: 11

The mode is the value that occurs the most. In this data set the mode is 8.

The mean, median and mode are: mean: ; median: ; mode: .

The data set is already ordered.

Since there is an even number of values in this data set the median lies between the fourth and fifth numbers:

The mode is the value that occurs the most. In this data set the mode is 24.

The mean, median and mode are: mean: ; median: ; mode: .

We first need to order the data set: .

Since there is an even number of values in this data set the median lies between the fifth and sixth numbers:

The mode is the value that occurs the most. In this data set the mode is 11.

Therefore the mean, median and mode are: mean: ; median: ; mode: .

We first need to order the data set:

Since there is an odd number of values in this data set the median lies at the fourth number: 31

The mode is the value that occurs the most. In this data set the mode is 31.

The mean, median and mode are: mean: ; median: ; mode: none.

The ages of runners of the Comrades Marathon were recorded:

Calculate the mean, median and modal age.

We first need to order the data set:

Since there is an odd number of values in this data set the median lies at the eighth number: 38.

The mode is the value that occurs the most. In this data set there are two modes: 33 and 42.

Therefore the mean, median and modal ages are: mean: ; median ; mode and .

A group of 10 friends each have some stones. They work out that the mean number of stones they have is 6. Then 7 friends leave with an unknown number () of stones. The remaining 3 friends work out that the mean number of stones they have left is .

When the 7 friends left, how many stones did they take with them?

If the mean number of stones the group originally had was 6 then the total number of stones must have been:

We are then told that 7 friends leave and thereafter the mean number of stones left is . Now we can work out the remaining number of stones.

Now we can calculate how many stones were taken by the 7 friends who left the group.

A group of 9 friends each have some coins. They work out that the mean number of coins they have is 4. Then 5 friends leave with an unknown number () of coins. The remaining 4 friends work out that the mean number of coins they have left is .

When the 5 friends left, how many coins did they take with them?

If the mean number of coins the group originally had was 4 then the total number of coins must have been:

We are then told that 5 friends leave and thereafter the mean number of coins left is . Let us work out the remaining number of coins.

Now we can calculate how many coins were taken by the 5 friends who left the group.

A group of 9 friends each have some marbles. They work out that the mean number of marbles they have is 3. Then 3 friends leave with an unknown number () of marbles. The remaining 6 friends work out that the mean number of marbles they have left is .

When the 3 friends left, how many marbles did they take with them?

If the mean number of marbles the group originally had was 3 then the total number of marbles must have been:

We are then told that 3 friends leave and thereafter the mean number of marbles left is . Let us work out the remaining number of marbles.

Now we can calculate how many marbles were taken by the 3 friends who left the group.

In the first of a series of jars, there is sweet. In the second jar, there are sweets. The mean number of sweets in the first two jars is .

If the mean number of sweets in the first three jars is , how many sweets are there in the third jar?

Let be the number of sweets in the third jar:

If the mean number of sweets in the first four jars is , how many sweets are there in the fourth jar?

Let be the number of sweets in the fourth jar:

Find a set of five ages for which the mean age is , the modal age is and the median age is years.

Let the five different ages be and . Therefore the mean is:

The median value is at position , therefore .

The mode is the age that occurs most often. We have 5 ages to work with and we know one of the ages is 3 (from the median). So the ordered data set is: (remember that we always calculate mean, mode and median using the ordered data set). We are told that the mode is 2. Looking at the ordered data set we see that either or must be 2 ( and cannot be 2 as that would make the data set unordered). However, if only one of these values is 2 then the mode will not be 2. Therefore .

So we can now update our calculation of the mean:

and can be any numbers that add up to and are not the same (if they were the same then the mode would not be 2), so and or and or and , etc.

Possible data sets:

Note that the set of ages must be ordered, the median value must be and there must be ages of .

Four friends each have some marbles. They work out that the mean number of marbles they have is . One friend leaves with marbles. How many marbles do the remaining friends have together?

Let the number of marbles per friend be and .

One friend leaves:

Therefore the remaining friends have marbles.

10.3 Grouping data

A common way of handling continuous quantitative data is to subdivide the full range of values into a few sub-ranges. By assigning each continuous value to the sub-range or class within which it falls, the data set changes from continuous to discrete.

Grouping is done by defining a set of ranges and then counting how many of the data fall inside each range. The sub-ranges must not overlap and must cover the entire range of the data set.

One way of visualising grouped data is as a histogram. A histogram is a collection of rectangles, where the base of a rectangle (on the -axis) covers the values in the range associated with it, and the height of a rectangle corresponds to the number of values in its range.

The following video explains how to group data.

Video: 2GMP

Worked example 9: Groups and histograms

The heights in centimetres of learners are given below.

Group the data into the following ranges and draw a histogram of the grouped data:

(Note that the ranges do not overlap since each one starts where the previous one ended.)

Count the number of values in each range

Range

Count

Draw the histogram

Since there are ranges, the histogram will have rectangles. The base of each rectangle is defined by its range. The height of each rectangle is determined by the count in its range.

The histogram makes it easy to see in which range most of the heights are located and provides an overview of the distribution of the values in the data set.

Exercise 10.3

A group of learners count the number of playing cards they each have. This is a histogram describing the data they collected:

Count the number of playing cards in the following range:

From the graph the answer is: 1

From the histogram, we arrive at our answer by reading the height of the specified interval from the histogram.

A group of learners count the number of stones they each have. This is a histogram describing the data they collected:

Count the number of stones in the following range:

From the graph the answer is: 1

From the histogram, we arrive at our answer by reading the height of the specified interval from the histogram.

A group of learners count the number of playing cards they each have. This is the data they collect:

Count the number of learners who have from up to playing cards. In other words, how many learners have playing cards in the following range: ? It may be helpful for you to draw a histogram in order to answer the question.

Firstly we sort the table into sequential order, starting with the smallest value.

Secondly, we draw a histogram of the data:

From the histogram you can see that the number of learners with playing cards in the range: is 3.

A group of learners count the number of stones they each have. This is the data they collect:

Count the number of learners who have from up to stones. In other words, how many learners have stones in the following range: ? It may be helpful for you to draw a histogram in order to answer the question.

Firstly we sort the table into sequential order, starting with the smallest value.

Secondly, we draw a histogram of the data:

From the histogram you can see that the number of learners with stones in the range: is 5.

A group of 20 learners count the number of stones they each have. The learners draw a histogram describing the data they collected. However, they have made a mistake in drawing the histogram.

The data set below shows the correct information for the number of stones the learners have. Each value represents the number of stones for one learner.

Help them figure out which column in the histogram is incorrect.

We first need to order the data:

Using the ordered data set we can group the data and draw the correct histogram:

The column with the error in it was: E.

The learners used the incorrect value of 0, when the correct value is .

A group of 20 learners count the number of stones they each have. The learners draw a histogram describing the data they collected. However, they have made a mistake in drawing the histogram.

The data set below shows the correct information for the number of stones the learners have. Each value represents the number of stones for one learner.

Help them figure out which column in the histogram is incorrect.

We first need to order the data:

Using the ordered data set we can group the data and draw the correct histogram:

The column with the error in it was: B.

The learners used the incorrect value of 5, when the correct value is .

A group of learners count the number of sweets they each have. This is a histogram describing the data they collected:

A cat jumps onto the table, and all their notes land on the floor, mixed up, by accident!

Help them find which of the following data sets match the above histogram:

Data Set A

Data Set B

Data Set C

The correct answer is: Data Set C

A group of learners count the number of stones they each have. This is a histogram describing the data they collected:

A cleaner knocks over their table, and all their notes land on the floor, mixed up, by accident!

Help them find which of the following data sets match the above histogram:

Data Set A

Data Set B

Data Set C

The correct answer is: Data Set C

A class experiment was conducted and learners were asked to guess the number of sweets in a jar. The following guesses were recorded:

Draw up a grouped frequency table using the intervals , , , and .

Group

Frequency

Draw the histogram corresponding to the frequency table of the grouped data.

Measures of central tendency

With grouped data our estimates of central tendency will change because we lose some information when we place each value in a range. If all we have to work with is the grouped data, we do not know the measured values to the same accuracy as before. The best we can do is to assume that values are grouped at the centre of each range.

Looking back to the previous worked example, we started with this data set of learners' heights.

Note that the data are sorted.

The mean of these data is and the median is . The mode is , but remember that there are problems with computing the mode of continuous quantitative data.

After grouping the data, we now have the data set shown below. Note that each value is placed at the centre of its range and that the number of times that each value is repeated corresponds exactly to the counts in each range.

The grouping changes the measures of central tendency since each datum is treated as if it occurred at the centre of the range in which it was placed.

The mean is now , the median and the mode is . This is actually a better estimate of the mode, since the grouping showed in which range the learners' heights were clustered.

We can also just give the modal group and the median group for grouped data. The modal group is the group that has the highest number of data values. The median group is the central group when the groups are arranged in order.

Exercise 10.4

Consider the following grouped data and calculate the mean, the modal group and the median group.

Mass ()

Count

To find the mean we use the middle value for each group. The count then tells us how many times that value occurs in the data set. Therefore the mean is:

The modal group is the group with the highest number of data values. This is with 15 data values.

The median group is the central group. There are 5 groups and so the central group is the third one: .

Mean: ; Modal group: ; Median group: .

Find the mean, the modal group and the median group in this data set of how much time people needed to complete a game.

Time (s)

Count

To find the mean we use the middle value for each group. The count then tells us how many times that value occurs in the data set. Therefore the mean is:

The modal group is the group with the highest number of data values. This is with 26 data values.

The median group is the central group. There are 7 groups and so the central group is the fourth one: .

Mean: ; Modal group: ; Median group: .

The histogram below shows the number of passengers that travel in Alfred's minibus taxi per week.

Calculate:

the modal interval

The modal interval is the interval with the highest number of data values. For this data set it is: with 16 values.

the total number of passengers to travel in Alfred's taxi

We add up the counts in each group and then multiply these counts with the central value for each group: .

an estimate of the mean

There are 48 values in the data set. Therefore the mean is .

an estimate of the median

We are looking for an estimate of the median rather than the median group here. In this case we note that there are 48 data values in the data set. Therefore the median will lie between the and values.

We note that 22 values in the first 3 groups and 38 values in the first four groups so the median must lie in the fourth group. Therefore we can estimate the median as the middle value of the fourth group: .

if it is estimated that every passenger travelled an average distance of , how much money would Alfred have made if he charged per km?

.

10.4 Measures of dispersion

The central tendency is not the only interesting or useful information about a data set. The two data sets illustrated below have the same mean (), but have different spreads around the mean. Each circle represents one value from the data set (or one datum).

Dispersion is a general term for different statistics that describe how values are distributed around the centre. In this section we will look at measures of dispersion.

Range

Range

The range of a data set is the difference between the maximum and minimum values in the set.

The most straightforward measure of dispersion is the range. The range simply tells us how far apart the largest and smallest values in a data set are. The range is very sensitive to outliers.

Worked example 10: Range

Find the range of the following data set:

What would happen if we removed the first value from the set?

Determine the range

The smallest value in the data set is and the largest value is .

The range is

Remove the first value

If the first value, , were to be removed from the set, the minimum value would be . This means that the range would change to . is not typical of the other values. It is an outlier and has a big influence on the range.

Percentiles

Percentile

The percentile is the value, , that divides a data set into two parts, such that percent of the values in the data set are less than and percent of the values are greater than . Percentiles can lie in the range .

To understand percentiles properly, we need to distinguish between different aspects of a datum: its value, its rank and its percentile:

The value of a datum is what we measured and recorded during an experiment or survey.

The rank of a datum is its position in the sorted data set (for example, first, second, third, and so on).

The percentile at which a particular datum is, tells us what percentage of the values in the full data set are less than this datum.

The table below summarises the value, rank and percentile of the data set:

Value

Rank

Percentile

As an example, is at the percentile since there are values less than and values greater than .

In general, the formula for finding the percentile in an ordered data set with values is

This gives us the rank, , of the percentile. To find the value of the percentile, we have to count from the first value in the ordered data set up to the value.

Sometimes the rank will not be an integer. This means that the percentile lies between two values in the data set. The convention is to take the value halfway between the two values indicated by the rank.

The figure below shows the relationship between rank and percentile graphically. We have already encountered three percentiles in this chapter: the median ( percentile), the minimum ( percentile) and the maximum (). The median is defined as the value halfway in a sorted data set.

Worked example 11: Using the percentile formula

Determine the minimum, maximum and median values of the following data set using the percentile formula.

Sort the values in the data set

Before we can use the rank to find values in the data set, we always have to order the values from the smallest to the largest. The sorted data set is

Find the minimum

We already know that the minimum value is the first value in the ordered data set. We will now confirm that the percentile formula gives the same answer. The minimum is equivalent to the percentile. According to the percentile formula the rank, , of the percentile in a data set with values is:

This confirms that the minimum value is the first value in the list, namely .

Find the maximum

We already know that the maximum value is the last value in the ordered data set. The maximum is also equivalent to the percentile. Using the percentile formula with and , we find the rank of the maximum value is:

This confirms that the maximum value is the last (the ninth) value in the list, namely .

Find the median

The median is equivalent to the percentile. Using the percentile formula with and , we find the rank of the median value is:

This shows that the median is in the middle (at the fifth position) of the ordered data set. Therefore the median value is .

Quartiles

The quartiles are the three data values that divide an ordered data set into four groups, where each group contains an equal number of data values. The median ( percentile) is the second quartile (). The percentile is also called the first or lower quartile (). The percentile is also called the third or upper quartile ().

Worked example 12: Quartiles

Determine the quartiles of the following data set:

Sort the data set

Find the ranks of the quartiles

Using the percentile formula with , we can find the rank of the , and percentiles:

Find the values of the quartiles

Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the percentile the rank is , which is between the third and fourth values. Since both these values are equal to , the percentile is .

For the percentile (the median) the rank is , meaning halfway between the sixth and seventh values. The sixth value is and the seventh value is , which means that the median is . For the percentile the rank is , meaning between the ninth and tenth values. Therefore the percentile is .

Deciles

The deciles are the nine data values that divide an ordered data set into ten groups, where each group contains an equal number of data values.

For example, consider the ordered data set:

The nine deciles are: .

Percentiles for grouped data

In grouped data, the percentiles will lie somewhere inside a range, rather than at a specific value. To find the range in which a percentile lies, we still use the percentile formula to determine the rank of the percentile and then find the range within which that rank is.

Worked example 13: Percentiles in grouped data

The mathematics marks of grade learners at a school have been collected. The data are presented in the following table:

Percentage mark

Number of learners

2

5

18

22

18

13

12

10

Calculate the mean of this grouped data set.

In which intervals are the quartiles of the data set?

In which interval is the percentile of the data set?

Calculate the mean

Since we are given grouped data rather than the original ungrouped data, the best we can do is approximate the mean as if all the learners in each interval were located at the central value of the interval.

Find the quartiles

Since the data have been grouped, they have also already been sorted. Using the percentile formula and the fact that there are learners, we can find the rank of the , and percentiles as

Now we need to find in which ranges each of these ranks lie.

For the lower quartile, we have that there are learners in the first two ranges combined and learners in the first three ranges combined. Since , this means the lower quartile lies somewhere in the third range: .

For the second quartile (the median), we have that there are learners in the first four ranges combined. Since , this means that the median lies somewhere in the fifth range: .

For the upper quartile, we have that there are learners in the first five ranges combined and learners in the first six ranges combined. Since , this means that the upper quartile lies somewhere in the sixth range: .

Find the percentile

Using the same method as for the quartiles, we first find the rank of the percentile.

Now we have to find the range in which this rank lies. Since there are learners in the first ranges combined and learners in the first ranges combined, the percentile lies in the fourth range:

Ranges

We define data ranges in terms of percentiles. We have already encountered the full data range, which is simply the difference between the and the percentile (that is, between the maximum and minimum values in the data set).

Interquartile range

The interquartile range is a measure of dispersion, which is calculated by subtracting the first quartile () from the third quartile (). This gives the range of the middle half of the data set.

Semi interquartile range

The semi interquartile range is half of the interquartile range.

Exercise 10.5

A group of learners count the number of sweets they each have. This is the data they collect:

Calculate the range of values in the data set.

We first need to order the data set:

Next we find the maximum value in the data set:

Then we find the minimum value in the data set:

Finally, we calculate the range of the data set:

A group of learners count the number of playing cards they each have. This is the data they collect:

Calculate the range of values in the data set.

We first need to order the data set:

Next we find the maximum value in the data set:

Then we find the minimum value in the data set:

Finally, we calculate the range of the data set:

Find the range of the data set

The data set is already ordered.

Firstly, we find the maximum value in the data set:

Secondly, we find the minimum value in the data set:

Finally, we calculate the range of the data set:

What are the quartiles of this data set?

We first order the data set.

Next we find the ranks of the quartiles. Using the percentile formula with , we can find the rank of the , and percentiles:

Find the values of the quartiles. Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the percentile the rank is , which is between the third and fourth values. Therefore the percentile is .

For the percentile (the median) the rank is , meaning halfway between the sixth and seventh values. Therefore the median is . For the percentile the rank is , meaning between the ninth and tenth values. Therefore the percentile is .

Therefore we get the following values for the quartiles: ; ; .

A class of learners writes a test and the results are as follows:

Find the range, quartiles and the interquartile range.

The data set is ordered.

The range is:

To find the quartiles we start by finding the ranks of the quartiles. Using the percentile formula with , we can find the rank of the , and percentiles:

Find the values of the quartiles. Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the percentile the rank is , which is between the third and fourth values. Therefore the percentile is .

For the percentile (the median) the rank is , meaning halfway between the sixth and seventh values. Therefore the median is . For the percentile the rank is , meaning between the ninth and tenth values. Therefore the percentile is .

Therefore we get the following values for the quartiles: ; ; .

Interquartile range:

Three sets of data are given:

For each data set find:

the range

All three data sets are ordered. To find the range we subtract the minimum value from the maximum value. Doing so for each data set gives the following values for the range.

Data set 1:

Data set 2:

Data set 3:

the lower quartile

For each data set . Therefore the rank of the percentile is the same for each data set: . Therefore for each data set the lower quartile lies between the second and third values.

The lower quartile for each data set is:

Data set 1:

Data set 2:

Data set 3:

the median

For each data set . Therefore the rank of the percentile is the same for each data set: . Therefore for each data set the median is the fourth value.

The median for each data set is:

Data set 1:

Data set 2:

Data set 3:

the upper quartile

For each data set . Therefore the rank of the percentile is the same for each data set: . Therefore for each data set the lower quartile lies between the fifth and sixth values.

The upper quartile for each data set is:

Data set 1:

Data set 2:

Data set 3:

the interquartile range

The interquartile range is calculated by subtracting the lower quartile from the upper quartile.

Data set 1:

Data set 2:

Data set 3:

the semi-interquartile range

The semi-interquartile range is half the interquartile range.

Data set 1:

Data set 2:

Data set 3:

10.5 Five number summary

A common way of summarising the overall data set is with the five number summary and the box-and-whisker plot. These two represent exactly the same information, numerically in the case of the five number summary and graphically in the case of the box-and-whisker plot.

The five number summary consists of the minimum value, the maximum value and the three quartiles. Another way of saying this is that the five number summary consists of the following percentiles: , , , , .

The box-and-whisker plot shows these five percentiles as in the figure below. The box shows the interquartile range (the distance between and ). A line inside the box shows the median. The lines extending outside the box (the whiskers) show where the minimum and maximum values lie. This graph can also be drawn horizontally.

This video explains how to draw a box-and-whisker plot for a data set.

Video: 2GNC

Wikihow shows a summary with short animations of how to make a box and whisker plot.

Worked example 14: Five number summary

Draw a box and whisker diagram for the following data set:

Determine the minimum and maximum

Since the data set is already sorted, we can read off the minimum as the first value () and the maximum as the last value ().

Determine the quartiles

There are values in the data set. Using the percentile formula, we can determine that the median lies between the sixth and seventh values, making it:

The first quartile lies between the third and fourth values, making it:

The third quartile lies between the ninth and tenth values, making it:

This provides the five number summary of the data set and allows us to draw the following box-and-whisker plot.

Exercise 10.6

Lisa is working in a computer store. She sells the following number of computers each month:

Give the five number summary and box-and-whisker plot of Lisa's sales.

We first order the data set.

Now we can read off the minimum as the first value () and the maximum as the last value ().

Next we need to determine the quartiles.

There are values in the data set. Using the percentile formula, we can determine that the median lies between the sixth and seventh values, making it:

The first quartile lies between the third and fourth values, making it:

The third quartile lies between the ninth and tenth values, making it:

This provides the five number summary of the data set and allows us to draw the following box-and-whisker plot.

Five number summary:

Minimum:

:

Median:

:

Maximum:

Box-and-whisker plot:

Zithulele works as a telesales person. He keeps a record of the number of sales he makes each month. The data below show how much he sells each month.

Give the five number summary and box-and-whisker plot of Zithulele's sales.

We first order the data set.

Now we can read off the minimum as the first value () and the maximum as the last value ().

Next we need to determine the quartiles.

There are values in the data set. Using the percentile formula, we can determine that the median lies between the sixth and seventh values, making it:

The first quartile lies between the third and fourth values, making it:

The third quartile lies between the ninth and tenth values, making it:

The five number summary is:

Minimum:

:

Median:

:

Maximum:

The box and whisker plot is:

Nombusa has worked as a florist for nine months. She sold the following number of wedding bouquets:

Give the five number summary of Nombusa's sales.

We first order the data set.

Now we can read off the minimum as the first value () and the maximum as the last value ().

Next we need to determine the quartiles.

There are values in the data set. Using the percentile formula, we can determine that the median lies at the fifth value, making it .

The first quartile lies at the third value, making it .

The third quartile lies at the seventh values, making it .

The five number summary is:

Minimum:

:

Median:

:

Maximum:

Determine the five number summary for each of the box-and-whisker plots below.

The box shows the interquartile range (the distance between and ). A line inside the box shows the median. The lines extending outside the box (the whiskers) show where the minimum and maximum values lie. Reading off the graph we obtain the following five number summary:

Minimum:

:

Median:

:

Maximum:

The box shows the interquartile range (the distance between and ). A line inside the box shows the median. The lines extending outside the box (the whiskers) show where the minimum and maximum values lie. Reading off the graph we obtain the following five number summary:

Minimum:

:

Median:

:

Maximum:

10.6 Chapter summary

Presentation: 2GNK

Data refer to the pieces of information that have been observed and recorded, from an experiment or a survey.

Quantitative data are data that can be written as numbers. Quantitative data can be discrete or continuous.

Qualitative data are data that cannot be written as numbers. There are two common types of qualitative data: categorical and anecdotal data.

The mean is the sum of a set of values divided by the number of values in the set.

The median of a data set is the value in the central position, when the data set has been arranged from the lowest to the highest value. If there are an odd number of data, the median will be equal to one of the values in the data set. If there are an even number of data, the median will lie half way between two values in the data set.

The mode of a data set is the value that occurs most often in the set.

An outlier is a value in the data set that is not typical of the rest of the set. It is usually a value that is much greater or much less than all the other values in the data set.

Continuous quantitative data can be grouped by dividing the full range of values into a few sub-ranges. By assigning each continuous value to the sub-range or class within which it falls, the data set changes from continuous to discrete.

Dispersion is a general term for different statistics that describe how values are distributed around the centre.

The range of a data set is the difference between the maximum and minimum values in the set.

The percentile is the value, , that divides a data set into two parts, such that of the values in the data set are less than and of the values are greater than . The general formula for finding the percentile in an ordered data set of values is

The quartiles are the three data values that divide an ordered data set into four groups, where each group contains an equal number of data values. The lower quartile is denoted , the median is and the upper quartile is .

The interquartile range is a measure of dispersion, which is calculated by subtracting the lower (first) quartile from the upper (third) quartile. This gives the range of the middle half of the data set.

The semi interquartile range is half of the interquartile range.

The five number summary consists of the minimum value, the maximum value and the three quartiles (, and ).

The box-and-whisker plot is a graphical representation of the five number summary.

Exercise 10.7

The following data set of heights was collected from a class of learners.

Categorise the data set.

This data set is a set of numbers and so must be quantitative.

This data set is continuous since it cannot be represented by integers.

Therefore the data set is quantitative continuous.

The following data set of sandwich spreads was collected from learners at lunch.

Categorise the data set.

This data set cannot be written as numbers and so must be qualitative.

This data set is categorical since it comes from a limited set of possibilities.

Therefore the data set is qualitative categorical.

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Calculate the mode of the following data set:

We first sort the data set: . The mode is the value that occurs most often in the data set.

Therefore the mode is: 10.

Calculate the median of the following data set:

We first need to order the data set:

.

Since there are an odd number of values in this data set (11) the median lies at the sixth place.

The median is: 10.

In a park, the tallest trees have heights (in metres):

Find the median of their heights.

We first need to order the data set:

.

Since there are an odd number of values in this data set (7) the median lies at the fourth place.

The median is: 44.

The learners in Ndeme's class have the following ages:

Find the mode of their ages.

We first sort the data set: . The mode is the value that occurs most often in the data set.

Therefore the mode is: 6.

A group of 7 friends each have some sweets. They work out that the mean number of sweets they have is 6. Then 4 friends leave with an unknown number () of sweets. The remaining 3 friends work out that the mean number of sweets they have left is .

When the 4 friends left, how many sweets did they take with them?

If the mean number of sweets the group originally had was 6 then the total number of sweets must have been:

We are then told that 4 friends leave and thereafter the mean number of sweets left is . Let us work out the remaining number of sweets.

Now we can calculate how many sweets were taken by the 4 friends who left the group.

A group of 10 friends each have some sweets. They work out that the mean number of sweets they have is 3. Then 5 friends leave with an unknown number () of sweets. The remaining 5 friends work out that the mean number of sweets they have left is 3.

When the 5 friends left, how many sweets did they take with them?

If the mean number of sweets the group originally had was 3 then the total number of sweets must have been:

We are then told that 5 friends leave and thereafter the mean number of sweets left is 3. Let us work out the remaining number of sweets.

Now we can calculate how many sweets were taken by the 5 friends who left the group.

Five data values are represented as follows: , with a mean of . Solve for .

Five data values are represented as follows: . Find the mean in terms of .

A group of learners count the number of marbles they each have. This is a histogram describing the data they collected:

Count the number of marbles in the following range:

From the graph the answer is: 1

From the histogram, we arrive at our answer by reading the height of the specified interval from the histogram.

A group of learners count the number of playing cards they each have. This is the data they collect:

Count the number of learners who have from up to playing cards. In other words, how many learners have playing cards in the following range: ? It may be helpful for you to draw a histogram in order to answer the question.

Firstly we sort the table into sequential order, starting with the smallest value.

Secondly, we draw a histogram of the data:

From the histogram you can see that the number of learners with playing cards in the range: is 4.

A group of learners count the number of coins they each have. This is the data they collect:

Count the number of learners who have from up to coins. In other words, how many learners have coins in the following range: ? It may be helpful for you to draw a histogram in order to answer the question.

Firstly we sort the table into sequential order, starting with the smallest value.

Secondly, we draw a histogram of the data:

From the histogram you can see that the number of learners with coins in the range: is 3.

A group of 20 learners count the number of playing cards they each have. The learners draw a histogram describing the data they collected. However, they have made a mistake in drawing the histogram.

The data set below shows the correct information for the number of playing cards the learners have. Each value represents the number of playing cards for one learner.

Help them figure out which column in the histogram is incorrect.

We first need to order the data:

Using the ordered data set we can group the data and draw the correct histogram:

The column with the error in it was: C.

The learners used the incorrect value of 3, when the correct value is .

A group of 10 learners count the number of sweets they each have. The learners draw a histogram describing the data they collected. However, they have made a mistake in drawing the histogram.

The data set below shows the correct information for the number of sweets the learners have. Each value represents the number of sweets for one learner.

Help them figure out which column in the histogram is incorrect.

We first need to order the data:

Using the ordered data set we can group the data and draw the correct histogram:

The column with the error in it was: E.

The learners used the incorrect value of 9, when the correct value is .

A group of learners count the number of sweets they each have. This is a histogram describing the data they collected:

A cleaner knocks over their table, and all their notes land on the floor, mixed up, by accident!

Help them find which of the following data sets match the above histogram:

Data set A

Data set B

Data set C

In order to determine which data set is correct we need to order each data set:

Data set A

Data set B

Data set C

Now we can group the data for each data set and compare the grouped data to the histogram. Doing so we find that data set C is the correct data set.

A group of learners count the number of marbles they each have. This is a histogram describing the data they collected:

A cat jumps onto the table, and all their notes land on the floor, mixed up, by accident!

Help them find which of the following data sets match the above histogram:

Data set A

Data set B

Data set C

In order to determine which data set is correct we need to order each data set:

Data set A

Data set B

Data set C

Now we can group the data for each data set and compare the grouped data to the histogram. Doing so we find that data set B is the correct data set.

A group of learners count the number of marbles they each have. This is the data they collect:

Calculate the range of values in the data set.

We need to order the data set:

Now we find the maximum value in the data set:

Next we find the minimum value in the data set:

Finally, we calculate the range of the data set.

A group of learners count the number of sweets they each have. This is the data they collect:

Calculate the range of values in the data set.

We first need to order the data set:

Next we find the maximum value in the data set.

Then we find the minimum value in the data set.

Finally, we calculate the range of the data set.

An engineering company has designed two different types of engines for motorbikes. The two different motorbikes are tested for the time (in seconds) it takes for them to accelerate from to .

Test

1

2

3

4

5

6

7

8

9

10

Bike 1

Bike 2

Which measure of central tendency should be used for this information?

Mean and mode. The mean will give us the average acceleration time, while the mode will give us the time that is most often obtained.

If we used the median we would not get any useful information as all the median tells us is what the central value is. The mean and mode provide more information about the data set as a whole.

Calculate the measure of central tendency that you chose in the previous question, for each motorbike.

We first sort the data.

Bike 1: .

Bike 2: .

Next we can calculate the mean for each bike:

For bike 1 the mean is and there is no mode, because there is no value that occurs more than once.

For bike 2 the mean is and there are two modes, and .

Which motorbike would you choose based on this information? Take note of the accuracy of the numbers from each set of tests.

It would be difficult to choose. Although bike 1 appears to do better than bike 2 from the mean, the data for bike 2 is less accurate than that for bike 1 (it only has 1 decimal place). If we were to calculate the mean for bike 1 using only 1 decimal place we would get . This would make bike 2 better. Also bike 2 produces more consistent numbers. So bike 2 would likely be a good choice, but more information or more accurate information should be obtained.

In a traffic survey, a random sample of motorists were asked the distance they drove to work daily. This information is shown in the table below.

Distance ()

Count

Find the approximate mean of the data.

To find the approximate mean we need to use the central value from each group. We are told that 50 motorists were surveyed and so the total number of data values is 50.

What percentage of drivers had a distance of

less than or equal to ?

more than ?

between and ?

The first three groups all drive less than or equal to . We can add up the number of drivers in these three groups and then divide this by the total number of drivers to find the percentage of drivers: .

The last three groups all drive more than . We can add up the number of drivers in these three groups and then divide this by the total number of drivers to find the percentage of drivers: .

The middle three groups fall into this range. We can add up the number of drivers in these three groups and then divide this by the total number of drivers to find the percentage of drivers: .

Note that the three percentages we have just calculated all add up to %.

Draw a histogram to represent the data.

We are given the groupings and the counts for each group. So we can draw the following histogram to represent the data:

A company wanted to evaluate the training programme in its factory. They gave the same task to trained and untrained employees and timed each one in seconds.

Trained

Untrained

Find the medians and quartiles for both sets of data.

First order the data sets for both trained and untrained employees.

Trained: .

Untrained: .

There are values in each data set.

Using the percentile formula with , we can find the rank of the , and percentiles:

For the percentile the rank is , which is between the fourth and fifth values. For the percentile (the median) the rank is . Therefore the median lies at the eighth value. For the percentile the rank is , meaning between the eleventh and values.

For the trained employees we get:

percentile: ; median: 129; percentile: .

For the untrained employees we get:

percentile: ; median: 144; percentile: .

Find the interquartile range for both sets of data.

Interquartile range for the trained employees: .

Interquartile range for the untrained employees: .

Comment on the results.

The median of the untrained employees is higher than that of the trained employees. Also the untrained employees have a larger interquartile range than the trained employees. There is some evidence to suggest that the training programme may be working.

Draw a box-and-whisker diagram for each data set to illustrate the five number summary.

A box-and-whisker plot shows the five number summary. The box shows the interquartile range (the distance between and ). A line inside the box shows the median. The lines extending outside the box (the whiskers) show where the minimum and maximum values lie.

Trained employees:

Untrained employees:

A small firm employs people. The annual salaries of the employees are:

Find the mean of these salaries.

Find the mode.

The mode is (this value occurs 3 times in the data set).

Find the median.

First order the data. To make the numbers easier to work with we will divide each one by .

The ordered set is .

The median is at position 5 and is .

Of these three figures, which would you use for negotiating salary increases if you were a trade union official? Why?

Either the mode or the median.The mean is skewed (shifted) by the one salary of . The mode gives us a better estimate of what the employees are actually earning. The median also gives us a fairly accurate representation of what the employees are earning.

The stem-and-leaf diagram below indicates the pulse rate per minute of ten Grade 10 learners.

Key: .

Determine the mean and the range of the data.

The data set is .

The mean and range are and 27 respectively.

Give the five-number summary and create a box-and-whiskers plot for the data.

The five-number summary is: .

Using this we can draw the box-and-whisker plot.

The following is a list of data:

In each separate case, determine the value of if the:

range =

The set is: . We have ordered all the numbers and then added on the end as we do not know the value of .

We are told that the range must be equal to 16. If the range would be . Therefore and so must be the maximum value.

mode =

is already the mode if we exclude , to maintain this is any integer with .

median =

First consider the set without : . The median in this set is 5 (there is an odd number of values in the set and the median lies at position 4).

Next we consider the full set: There is an even number of values (8) in the full set. Therefore the median must lie between the fourth and fifth values.

Now we need to think about where could fit into the set. could be the fourth value, the fifth value or somewhere else in the set. If is either the fourth or fifth value we will get the same median.

First try the case where is the fourth or fifth value:

Next we check the case where is not the fourth or fifth value. In this case the median is . Therefore we can say that .

mean =

box-and whiskers plot

In part a we found that if the range is 8, then . The range here is 8, therefore .

Therefore we will use the median to help us find . The median on the box-and-whiskers plot is . From our reasoning in part c we know that this means is either the fourth or fifth value. So we can calculate as follows:

Write down one list of numbers that satisfies the box-and-whisker plot below:

From the box-and-whisker plot we get the five number summary: . Note that the third quartile is also the maximum value in this case.

From this we can state that the data set must have a minimum value of 2 and a maximum value of 10.

The data set can contain any numbers in the range such that the first quartile is 5, the median is 7 and the third quartile is 10. There is also no restriction on how many values are in the data set.

One possible set that satisfies this set of numbers is . You can check that this set works by calculating the quartiles.

Given (which represents the golden ratio) to 20 decimal places:

For the first 20 decimal digits of , determine the:

median

mode

mean

The ordered set is:

The mode is .

If the mean of the first 21 decimal digits of is determine the decimal digit.

Below is a box-and-whisker plot of the - decimal digits. Write down one list of numbers that satisfies this box-and-whisker plot.

From the box-and-whisker plot we get the five number summary: . Note that the third quartile is also the maximum value in this case.

From this we can state that the data set must have a minimum value of 3 and a maximum value of 8.

The data set can contain any numbers in the range such that the first quartile is 4, the median is 5 and the third quartile is 8. However, we know that the data set consists of the - decimal digits of and so the data set must contain 7 values.

The median will be at the fourth position and so the fourth number in the set is 5. The first quartile will lie between the second and third values while the third quartile will lie between the fifth and sixth values.

Let the data set be:

The first quartile is:

and can be any integers that add up to 8. However and must be greater than or equal to 3 and less than or equal to 5. Therefore the possible values are: 3 and 5 or 4 and 4.

The third quartile is:

and can be any integers that add up to 16. However and must be greater than or equal to 5 and less than or equal to 8. Therefore the only possible values are: 8 and 8.

There are two possible sets: or .

There are 14 men working in a factory. Their ages are :

Write down the five number summary.

The five number summary is:

If 3 men had to be retrenched, but the median had to stay the same, show the ages of the men you would retrench.

Retrenching men will leave men. For an odd numbered set the median must be the same as one of the ages. None of the men are years.

Therefore no men can be retrenched to keep the median the same.

Find the mean age of the men in the factory using the original data.

The example below shows a comparison of the amount of dirt removed by four brands of detergents (brands to ).

Which brand has the biggest range, and what is this range?

A:

B:

C:

D:

B has the biggest range. The range is 9.

For brand , what does the number represent?

represents the median.

Give the interquartile range for brand .

Which brand of detergent would you buy? Explain your answer.

We need to compare several values to help us decide. These values are shown in the table below.

Brand | Minimum value | Maximum value | Range | Interquartile range
A | 10 | 17 | 7 | 4
B | 12 | 21 | 9 | 3
C | 15 | 22 | 7 | 4
D | 11 | 18 | 7 | 4

From this we see that brand C has the highest minimum value. Brand B has the smallest interquartile range but the largest range. It is possible that the minimum value for brand B is an outlier which would make brand B a better choice than brand C.

Considering all the data available it would be hard to choose between brand C and brand B. We can however say that brands A and D are not very good choices as they both have low minimum and maximum values.

Since brand C does not have a potential outlier this might be the best brand to choose.

Chapter 11: Trigonometry

This chapter covers solving problems in two-dimensions using trigonometry.

Emphasise the value and importance of making sketches, where appropriate.

Prior to starting this chapter it may be appropriate to quickly revise the earlier content on trigonometry.

Trigonometry was developed in ancient civilisations to solve practical problems such as building construction and navigating by the stars. We will show that trigonometry can also be used to solve some other practical problems. We can use the trigonometric ratios to solve problems in two dimensions that involve right-angled triangles.

As revision the three trigonometric ratios can be defined for right-angled triangles as:

We will use these three ratios and the theorem of Pythagoras to help us solve two-dimensional problems.

11.1 Two-dimensional problems

In two-dimensional problems we will often refer to the angle of elevation and the angle of depression. To understand these two angles let us consider a person sailing alongside some cliffs. The person looks up and sees the top of the cliffs as shown below:

In this diagram is the angle of elevation.

Angle of elevation

The angle of elevation is the angle formed by the line of sight and the horizontal plane for an object above the horizontal plane.

In our diagram the line of sight is from the ship to the top of the cliffs. The horizontal plane is from the ship to the base of the cliffs. Also note the we can consider the cliffs to be a straight vertical line and so we have a right-angled triangle.

To understand the angle of depression let us now consider the same situation as above but instead our observer is standing on top of the cliffs looking down at the ship.

In this diagram is the angle of depression.

Angle of depression

The angle of depression is the angle formed by the line of sight and the horizontal plane for an object below the horizontal plane.

In our diagram the line of sight is from the top of the cliffs to the ship. The horizontal plane is from the top of the cliffs through . Note that this is parallel to the line between the base of the cliffs and the ship. lies directly above the ship. We can construct a vertical, perpendicular line to the horizontal plane at the point .

Finally we can compare the angle of elevation and the angle of depression. In the following diagram the line from the base of the cliffs to the ship is parallel to the line from the top of the cliffs to . The angle of elevation and the angle of depression are indicated. Notice that .

Figure 11.1: An inclinometer. Inclinometers can be used to measure angles of inclination and so can be used to determine the height of an object.

You can make an inclinometer to measure the angle of inclination of a tall building or tree. Once you have the angle of inclination you can determine the height of the building or tree.

In trigonometry the angle of inclination is the same as the angle of elevation.

Worked example 1: Flying a kite

Mandla flies a kite on a string at an inclination of °.

What is the height, , of the kite above the ground?

If Mandla's friend Sipho stands directly below the kite, calculate the distance, , between the two friends.

Make a sketch and identify the opposite and adjacent sides and the hypotenuse

Use given information and appropriate ratio to solve for and

Note that the third side of the triangle can also be calculated using the theorem of Pythagoras: .

Write the final answers

The kite is above the ground.

Mandla and Sipho are apart.

Worked example 2: Calculating angles

is a trapezium with , , and . Point on diagonal divides the diagonal such that . . Find .

Draw trapezium and label all given lengths on diagram. Indicate that

We will use and to find and . We can then add these two angles together to find .

Find the first angle,

The hypotenuse and opposite side are given for both triangles, therefore use the function.

In :

Use the theorem of Pythagoras to determine

In :

Find the second angle

In :

Calculate the sum of the angles

Another application is using trigonometry to find the height of a building. We could use a tape measure lowered from the roof, but this is impractical (and dangerous) for tall buildings. It is much more sensible to use trigonometry.

Worked example 3: Finding the height of a building

The given diagram shows a building of unknown height . We start at point and walk away from the building to point . Next we measure the angle of elevation from the ground to the top of the building, , and find that the angle is . Calculate the height of the building, correct to the nearest metre.

Identify the opposite and adjacent sides and the hypotenuse

We have a right-angled triangle and know the length of one side and an angle. We can therefore calculate the height of the building.

In :

Rearrange and solve for

Write final answer

The height of the building is .

Worked example 4: Angles of elevation and depression

A block of flats is away from a cellphone tower. Someone stands at . They measure the angle from to the top of the tower () to be (the angle of elevation). They then measure the angle from to the bottom of the tower () to be (the angle of depression).

What is the height of the cellphone tower (correct to the nearest metre)?

Note: diagram not drawn to scale

To determine height , first calculate lengths and

and are both right-angled triangles. In each of the triangles, the length is known. Therefore we can calculate the sides of the triangles.

Calculate

The length is given. is a rectangle so .

In :

Calculate

In :

Add the two heights to get the final answer

The height of the tower is: .

Worked example 5: Building plan

Mr Nkosi has a garage at his house and he decides to add a corrugated iron roof to the side of the garage. The garage is high, and his sheet for the roof is long. If the angle of the roof is °, how high must he build the wall ? Give the answer correct to decimal place.

Identify opposite and adjacent sides and hypotenuse

is right-angled. The hypotenuse and an angle are known therefore we can calculate . The height of the wall is then the height of the garage minus .

Write the final answer

Mr Nkosi must build his wall to be high.

Exercise 11.1

A person stands at point , looking up at a bird sitting on the top of a building, point .

The height of the building is meters, the line of sight distance from point to the top of the building (point ) is meters, and the angle of elevation to the top of the building is .

Calculate the height of the building () as shown in the diagram below:

The height of the building is .

A person stands at point , looking up at a bird sitting on the top of a pole (point ).

The height of the pole is meters, point A is meters away from the foot of the pole, and the angle of elevation to the top of the pole is .

Calculate the height of the pole (), to the nearest metre.

The height of the pole is .

A boy flying a kite is standing from a point directly under the kite. If the kite's string is long, find the angle of elevation of the kite.

First draw a sketch:

We can use the cosine ratio to find the angle of elevation ():

The angle of elevation of the kite is °.

What is the angle of elevation of the sun when a tree tall casts a shadow long?

First draw a sketch:

We can use the tangent ratio to find the angle of elevation ():

The angle of elevation of the sun is °.

From a distance of , Susan looks up at the top of a lighthouse. The angle of elevation is °. Determine the height of the lighthouse to the nearest metre.

First draw a sketch:

We need to find . We can use the tangent ratio:

The height of the lighthouse is .

A ladder of length is resting against a wall, the ladder makes an angle ° to the wall. Find the distance between the wall and the base of the ladder to the nearest metre.

First draw a sketch:

Notice that we are given the angle that the ladder makes with the wall, not the angle that the ladder makes with the ground.

Now we can use the sine function to find :

The base of the ladder is away from the wall.

11.2 Chapter summary

Presentation: 2GPV

We can define three trigonometric ratios for right-angled triangles: sine (), cosine () and tangent ().

Trigonometry is used to help us solve problems in two dimensions that involve right-angled triangles, such as finding the height of a building.

The angle of elevation is the angle formed by the line of sight and the horizontal plane for an object above the horizontal plane.

The angle of depression is the angle formed by the line of sight and the horizontal plane for an object below the horizontal plane.

Exercise 11.2

A ladder of length is resting against a wall, the base of the ladder is from the wall. Find the angle between the wall and the ladder.

First draw a sketch:

Notice that we want to find the angle that the ladder makes with the wall, not the angle that the ladder makes with the ground.

Now we use :

The angle between the ladder and the wall is °.

Captain Jack is sailing towards a cliff with a height of .

The distance from the boat to the bottom of the cliff is . Calculate the angle of elevation from the boat to the top of the cliff (correct to the nearest degree).

First draw a sketch:

The angle of elevation is °.

If the boat sails closer to the cliff, what is the new angle of elevation from the boat to the top of the cliff?

We redraw the sketch with the new information:

The new distance from the boat to the cliffs is . The height of the cliffs has not changed.

The new angle of elevation is °.

Jim stands at point at the base of a telephone pole, looking up at a bird sitting on the top of another telephone pole (point ).

The height of each of the telephone poles is meters, and the angle of elevation from to the top of is .

Calculate the distance between the telephone poles () as shown in the diagram below:

The distance between the telephone poles is .

Alfred stands at point , looking up at a flag on a pole (point ).

Point is meters away from the bottom of the flag pole, the line of sight distance from point to the top of the flag pole (point ) is meters, and the angle of elevation to the top of the flag pole is .

Calculate the angle of elevation to the top of the flag pole () as shown in the diagram below:

The angle of elevation is .

A rugby player is trying to kick a ball through the poles. The rugby crossbar is high. The ball is placed from the poles. What is the minimum angle he needs to launch the ball to get it over the bar?

is the distance from the poles, ; BC is the crossbar height . The minimum angle is the angle of elevation.

Therefore he needs to kick the ball with a minimum angle of .

The escalator at a mall slopes at an angle of 30° and is long.

Through what height would a person be lifted by travelling on the escalator?

We note that we have the following right-angled triangle:

We can use the sine ratio to find the height:

A person travelling on the escalator would be lifted through a height of .

A ladder is metres long. It is leaning against the wall of a house and reaches metres up the wall.

Draw a sketch of the situation.

Calculate the angle which the ladder makes with the flat (level) ground.

The ladder makes an angle of ° with the ground.

Nandi is standing on level ground metres away from a tall tower. From her position, the angle of elevation of the top of the tower is °.

Draw a sketch of the situation.

What is the height of the tower?

We can use the tangent ratio to find the height of the tower.

The height of the tower is .

The top of a pole is anchored by a cable which makes an angle of degrees with the horizontal. What is the height of the pole?

The height of the pole is .

A ship's navigator observes a lighthouse on a cliff. According to the navigational charts the top of the lighthouse is 35 metres above sea level. She measures the angle of elevation of the top of the lighthouse to be °.

Ships have been advised to stay at least away from the shore. Is the ship safe?

First draw a diagram:

Therefore the ship is not safe.

Determine the perimeter of rectangle :

Using trigonometric ratios we can calculate and .

Therefore the perimeter is .

A rhombus has diagonals of lengths and . Calculate the sizes of its interior vertex angles.

There are four small right-angled triangles in the rhombus: , , and . Since the diagonals bisect each other and we are given the lengths of the diagonals, we know the lengths of the sides of the triangles:

We can calculate the two angles:

Now we note that there are two different interior angles. One of these angles is and the other is .

Therefore the two angles are ° and °

A rhombus has edge lengths of . Its acute interior vertex angles are both 70°. Calculate the lengths of both of its diagonals.

Therefore the two diagonals are and

A parallelogram has edge-lengths of and respectively, and an angle of 58° between them. Calculate the perpendicular distance between the two longer edges.

One of the angles of a rhombus with perimeter is °.

Find the lengths of the sides of the rhombus.

First draw a sketch:

The perimeter is found by adding each side together. All the sides are equal in length, therefore the perimeter .

Therefore the sides are all in length.

Find the length of both diagonals.

The diagonals of a rhombus bisect the angle, so working in one of the small triangles we can use trigonometric ratios to find :

By Pythagoras :

Since the diagonals bisect each other we know that the total length of each diagonal is either or , depending on which diagonal we examine.

The one diagonal is and the other diagonal is .

Upright sticks and the shadows they cast can be used to judge the sun's altitude in the sky (the angle the sun makes with the horizontal) and the heights of objects.

An upright stick, 1 metre tall, casts a shadow which is metres long. What is the altitude of the sun?

At the same time, the shadow of a building is found to be 47 metres long. What is the height of the building?

We know the angle that the sun makes with the horizontal and now we can use that to find the height of the building.

In the figure above is the angle that the sun makes with the horizontal.

The angle of elevation of a hot air balloon, climbing vertically, changes from 25 degrees at 11:00 am to 60 degrees at 11:02 am. The point of observation of the angle of elevation is situated 300 metres away from the take off point.

Draw a sketch of the situation.

Calculate the increase in height between 11:00 am and 11:02 am.

The difference is:

When the top, , of a mountain is viewed from point , from the ground, the angle of depression () is equal to 15°. When it is viewed from point on the ground, the angle of elevation () is equal to 10°. If the points and are on the same vertical line, find the height, , of the mountain. Round your answer to one decimal place.

The diagram below shows quadrilateral , with , , and angles and are right angles.

Find , correct to 2 decimal places.

Find the size of the angle marked , correct to one decimal place.

From a boat at sea (), the angle of elevation of the top of a lighthouse , on a cliff , is 27°.

The lighthouse is high and the cliff top is above sea level.

How far is the boat from the base of the cliff, to the nearest metre?

First draw a sketch:

The distance is equal to the height of the lighthouse, , plus the height of the cliffs, .

Chapter 12: Euclidean geometry

This chapter focuses on solving problems in Euclidean geometry and proving riders.

It must be explained that a single counter example can disprove a conjecture but numerous specific examples supporting a conjecture do not constitute a general proof.

To prove that a quadrilateral is one of the special quadrilaterals learners need to show that a unique property of that quadrilateral is true. For example to prove a quadrilateral is a parallelogram it is not enough to show that both pairs of sides are parallel, learners will also need to show that either the opposite angles are equal or both pairs of opposite sides are equal in length.

Geometry (from the Greek “geo” = earth and “metria” = measure) arose as the field of knowledge dealing with spatial relationships. Geometry can be split into Euclidean geometry and analytical geometry. Analytical geometry deals with space and shape using algebra and a coordinate system. Euclidean geometry deals with space and shape using a system of logical deductions.

12.1 Proofs and conjectures

We will now apply what we have learnt about geometry and the properties of polygons (in particular triangles and quadrilaterals) to prove some of these properties. We will also look at how we can prove a particular quadrilateral is one of the special quadrilaterals.

This video shows how to prove that the the diagonals of a rhombus are perpendicular.

Video: 2GQM

Worked example 1: Proving a quadrilateral is a parallelogram

In parallelogram , the bisectors of the angles (, , and ) have been constructed. You are also given , , , , , . Prove that is a parallelogram.

Use properties of the parallelogram to fill in on the diagram all equal sides and angles.

Prove that

In and ,

In and

Similarly, we can show that

First show . Then show .

Conclusion

Both pairs of opposite angles of are equal. Therefore is a parallelogram.

This video shows how to prove that the opposite angles of a parallelogram are equal.

Video: 2GQN

Exercise 12.1

In the diagram below, and bisect each other at . is the midpoint of , and is the midpoint of .

Prove is a parallelogram.

and bisect each other (given).

is a parallelogram (diagonals bisect each other).

Prove is a parallelogram.

is a parallelogram)

(mid-point, given)

( is a parallelogram)

is a parallelogram (two sides are parallel and equal)

Parallelogram and are shown below. Prove .

is a parallelogram. . Prove

Study the quadrilateral with opposite angles and angles carefully. Fill in the missing reasons and steps to prove that the quadrilateral is a parallelogram.

Here is the completed proof with the correct steps and reasons.

Study the quadrilateral with opposite angles and angles carefully. Fill in the missing reasons and steps to prove that the quadrilateral is a parallelogram.

Here is the completed proof with the correct steps and reasons.

Quadrilateral with sides and is given. You are also given that: and ; and . Prove that is a parallelogram.

Find the value of .

is a parallelogram (proved above).

and (opp s of m).

Therefore, .

Find the value of .

We can solve this problem in two ways: using the sum of angles in a triangle or using the sum of the interior angles in a quadrilateral.

Option 1: sum of angles in a triangle.

(sum of s in ).

We know that and that .

.

Option 2: sum of angles in a quadrilateral.

The sum of the interior 's in a quadrilateral is .

Quadrilateral with sides and is given. Also given is and ; and . Prove that is a parallelogram.

Determine the value of .

is a parallelogram, .

Opposite 's of a parallelogram are equal: and .

Therefore, .

Determine the value of .

We can solve this problem in two ways: using the sum of angles in a triangle or using the sum of interior angles in a quadrilateral.

Option 1: sum of angles in a triangle.

's in a

Now we know that and that .

.

Option 2: sum of interior angles in a quadrilateral.

The sum of the interior 's in a quadrilateral is .

In parallelogram , the bisectors of the angles have been constructed, indicated with the red lines below. You are also given , , , , and .

Prove that the quadrilateral is a parallelogram.

Note the diagram is drawn to scale.

Study the diagram below; it is not necessarily drawn to scale. Two triangles in the figure are congruent: . Additionally, . You need to prove that is a parallelogram.

Redraw the diagram and mark all given and known information:

Study the diagram below; it is not necessarily drawn to scale. Quadrilateral is a parallelogram and and have lengths and , respectively, as shown. You need to prove that .

Redraw the diagram and fill in all given and known information.

12.2 Chapter summary

Presentation: 2GR2

A quadrilateral is a closed shape consisting of four straight line segments.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Both pairs of opposite sides are equal in length.

Both pairs of opposite angles are equal.

Both diagonals bisect each other.

A rectangle is a parallelogram that has all four angles equal to

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are equal in length.

The diagonals bisect each other.

The diagonals are equal in length.

All interior angles are equal to

A rhombus is a parallelogram that has all four sides equal in length.

Both pairs of opposite sides are parallel.

All sides are equal in length.

Both pairs of opposite angles are equal.

The diagonals bisect each other at

The diagonals of a rhombus bisect both pairs of opposite angles.

A square is a rhombus that has all four interior angles equal to

Both pairs of opposite sides are parallel.

The diagonals bisect each other at

All interior angles are equal to

The diagonals are equal in length.

The diagonals bisect both pairs of interior opposite angles (i.e. all are )

A trapezium is a quadrilateral with one pair of opposite sides parallel.

A kite is a quadrilateral with two pairs of adjacent sides equal.

One pair of opposite angles are equal (the angles are between unequal sides).

The diagonal between equal sides bisects the other diagonal.

The diagonal between equal sides bisects the interior angles.

The diagonals intersect at

The mid-point theorem: The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

This video shows how to prove that the opposite sides of a parallelogram are equal.

Video: 2GR3

Exercise 12.2

is a rhombus with and . Prove is also a rhombus.

In and

(given rhombus , diagonal bisects )

(s opp equal sides)

similarly

and

but these are alternate interior s

and

is a parallelogram

(opp sides of m)

is a rhombus (all sides equal and two pairs of sides parallel)

is a parallelogram with diagonal . Given that , show that:

is a parallelogram

Given parallelogram with bisecting and bisecting .

Write all interior angles in terms of .

First number the angles:

Prove that is a parallelogram.

Given that , and , prove that:

bisects

First label the angles:

is a point on , in . is the mid-point of . is the mid-point of and is the mid-point of . .

Prove that is a parallelogram.

(given)

(given)

(Midpt Theorem)

(given)

(given)

(Midpt Theorem)

is a parallelogram (both pairs opp. sides parallel)

Prove that .

(given)

(given)

(Midpt Theorem)

(given)

In , , is the mid-point of and is the mid-point of .

Prove is the mid-point of .

If and the area of is , calculate the area of .

Prove that the area of will always be four times the area of , let and .

Given quadrilateral with sides and . Also given: and ; and . Complete the proof below to prove that is a parallelogram.

The completed proof looks like this:

Calculate the value of .

is a parallelogram, .

Opposite s of parallelogram are equal. and .

Therefore, .

Calculate the value of .

s in a

Now we know that and that .

.

Study the quadrilateral with opposite angles and angles carefully. Fill in the correct reasons or steps to prove that the quadrilateral is a parallelogram.

Study the quadrilateral with and carefully. Fill in the correct reasons or steps to prove that the quadrilateral is a parallelogram.

In parallelogram , the bisectors of the angles have been constructed, indicated with the red lines below. You are also given , , , , and .

Prove that the quadrilateral is a parallelogram.

Note the diagram is drawn to scale.

Redraw the diagram and mark all the known information:

Study the diagram below; it is not necessarily drawn to scale. Two triangles in the figure are congruent: . Additionally, . You need to prove that is a parallelogram.

Redraw the diagram and mark all known and given information:

Given the following diagram:

Show that is a parallelogram.

Show that is a parallelogram.

But and are diagonals of , is a parallelogram (diagonals bisect each other).

Prove that .

(proved above).

is a parallelogram. is a parallelogram. is a straight line. Prove that .

In the figure below , . is a parallelogram. Prove is a straight line.

We note that:

We also note that:

Now we can show that is congruent to :

Finally we can show that is a straight line:

Chapter 13: Measurements

Content covered in this chapter includes revision of volume and surface area for right-prisms and cylinders. This work is then extended to spheres, right pyramids and cones. Finally learners investigate the effects of multiplying any dimension by a constant factor .

Restrict pyramids to those with square or equilateral triangles as bases.

Composite figures must be included e.g. a square pyramids on top of a cube or two cones stuck together.

For revision of earlier grades content on surface area and perimeter you can have learners look up the regulations for the size of different sports fields and calculate the perimeter and surface area for different parts of the field.

Knowing how to calculate the surface area and volume of an object can be useful in many contexts, particularly when we need to know how much a task is going to cost or how much material is needed to create an object. Some examples of this are calculating the surface area of a container, to help us work out the cost of the material, or calculating the volume of a dam, so we know how much water the dam can hold.

This chapter examines the surface areas and volumes of three dimensional objects, otherwise known as solids. In order to work with these objects, you need to know how to calculate the surface area and perimeter of two dimensional shapes.

Figure 13.1: A tennis court. The position of each of the lines is carefully calculated to ensure that the area of the rectangles is the same anywhere in the world.

To revise how to calculate the area and perimeter of squares and rectangles you can watch the video below.

Video: 2GRM

13.1 Area of a polygon

Area

Area is the two dimensional space inside the boundary of a flat object. It is measured in square units.

The acre and the hectare are two common measurements used for the area of land. One hectare is about square kilometres and one acre is about square kilometres.

Name

Shape

Formula

Square

Rectangle

Triangle

Trapezium

Parallelogram

Circle

The video below shows some examples of calculations involving the area of a circle.

Video: 2GRN

Worked example 1: Finding the area of a polygon

Find the area of the following parallelogram:

Find the height

Find the area using the formula for a parallelogram

The following Phet simulation allows you to build different shapes and calculate the area and perimeter for the shapes: Phet: area builder.

Exercise 13.1

Find the area of each of the polygons below:

The radius is half the diameter, therefore the radius is .

We first need to work out the height using the theorem of Pythagoras:

Now we can calculate the area:

We first need to work out the height using the theorem of Pythagoras:

Now we can calculate the area:

We first need to construct the vertical (or perpendicular) height. For an isosceles triangle if we construct the perpendicular height between the two equal sides then this line will bisect the third side.

Now we can calculate the height using the theorem of Pythagoras:

Now we can calculate the area:

We first construct the vertical (perpendicular) height. For an equilateral triangle the perpendicular height will bisect the third side.

Now we can calculate the height using the theorem of Pythagoras:

Now we can calculate the area:

We first find the height using the theorem of Pythagoras:

Now we can calculate the area:

Find an expression for the area of this figure in terms of and . The circle has a radius of . Write your answer in expanded form (not factorised).

Find an expression for the area of this figure in terms of and . The height of the figure is , and two sides are labelled as and . Write your answer in expanded form (not factorised).

Find an expression for the area of this figure in terms of and . The circle has a radius of . Write your answer in expanded form (not factorised).

Find an expression for the area of this figure in terms of and . The height of the figure is , and two sides are labelled as and . Write your answer in expanded form (not factorised).

13.2 Right prisms and cylinders

Right prism

A right prism is a geometric solid that has a polygon as its base and vertical faces perpendicular to the base. The base and top surface are the same shape and size. It is called a “right” prism because the angles between the base and faces are right angles.

A triangular prism has a triangle as its base, a rectangular prism has a rectangle as its base, and a cube is a rectangular prism with all its sides of equal length. A cylinder has a circle as its base. Examples of right prisms and a cylinder are given below: a rectangular prism, a cube and a triangular prism.

Surface area of prisms and cylinders

Surface area

Surface area is the total area of the exposed or outer surfaces of a prism.

This is easier to understand if we imagine the prism to be a cardboard box that we can unfold. A solid that is unfolded like this is called a net. When a prism is unfolded into a net, we can clearly see each of its faces. In order to calculate the surface area of the prism, we can then simply calculate the area of each face, and add them all together.

For example, when a triangular prism is unfolded into a net, we can see that it has two faces that are triangles and three faces that are rectangles. To calculate the surface area of the prism, we find the area of each triangle and each rectangle, and add them together.

In the case of a cylinder the top and bottom faces are circles and the curved surface flattens into a rectangle with a length that is equal to the circumference of the circular base. To calculate the surface area we therefore find the area of the two circles and the rectangle and add them together.

Below are examples of right prisms and a cylinder that have been unfolded into nets:

Rectangular prism

Rectangular prism

A rectangular prism unfolded into a net is made up of six rectangles.

Cube

Cube

A cube unfolded into a net is made up of six identical squares.

Triangular prism

Triangular prism

A triangular prism unfolded into a net is made up of two triangles and three rectangles. The sum of the lengths of the rectangles is equal to the perimeter of the triangles.

Cylinder

Cylinder

A cylinder unfolded into a net is made up of two identical circles and a rectangle with a length equal to the circumference of the circles.

This video explains how we can unfold solids into nets.

Video: 2GS2

It may be useful to have some nets of the different polyhedra available for learners to see how they fold up to form the polyhedra.

You can download and print out nets of various polyhedra from senteacher to use in your classroom.

Worked example 2: Finding the surface area of a rectangular prism

Find the surface area of the following rectangular prism:

Sketch and label the net of the prism

Find the areas of the different shapes in the net

Find the sum of the areas of the faces

Write the final answer

The surface area of the rectangular prism is .

Worked example 3: Finding the surface area of a triangular prism

Find the surface area of the following triangular prism:

Sketch and label the net of the prism

Find the area of the different shapes in the net

To find the area of the rectangle, we need to calculate its length, which is equal to the perimeter of the triangles.

To find the perimeter of the triangle, we have to first find the length of its sides using the theorem of Pythagoras:

Find the sum of the areas of the faces

Write the final answer

The surface area of the triangular prism is .

Worked example 4: Finding the surface area of a cylinder

Find the surface area of the following cylinder (correct to 1 decimal place):

Sketch and label the net of the cylinder

Find the area of the different shapes in the net

Write the final answer

The surface area of the cylinder is .

Exercise 13.2

Calculate the surface area of the following prisms:

There are three different sized rectangles that make up the sides of this triangular prism. We need to find the area of each one of them. All of the rectangles have a height of 11 but each rectangle has a different base.

There are 4 rectangles and 2 squares that make up this rectangular prism. The square has a side length of 5. The rectangles have a base of 5 and a height of 11.

We first need to find the missing side of the triangle. We can do this using the theorem of Pythagoras.

Now we can find the area of the triangular prism:

If a litre of paint covers an area of , how much paint does a painter need to cover:

a rectangular swimming pool with dimensions (the inside walls and floor only);

We need to find the surface area of the pool. In this case we have a rectangular prism but with one rectangle missing (which would be the top of the pool).

The painter needs one litre of paint for every of area. So we must divide the surface area by 2 to find the total amount of paint needed. Therefore, the painter will need of paint (rounded up to the nearest litre).

the inside walls and floor of a circular reservoir with diameter and height .

We need to find the surface area of the reservoir. In this case we have a cylinder but with one circle missing (which would be the top of the reservoir).

We are given the diameter of the reservoir. The radius is half the diameter and so .

The painter needs one litre of paint for every of area. So we must divide the surface area by 2 to find the total amount of paint needed. Therefore, the painter will need of paint (rounded up to the nearest litre).

Volume of prisms and cylinders

Volume

Volume is the three dimensional space occupied by an object, or the contents of an object. It is measured in cubic units.

The volume of right prisms and cylinders is simply calculated by multiplying the area of the base of the solid by the height of the solid.

The video below shows several examples of calculating the volume of a right prism.

Video: 2GSB

Rectangular prism

Triangular prism

Cylinder

Worked example 5: Finding the volume of a cube

Find the volume of the following cube:

Find the area of the base

Multiply the area of the base by the height of the solid to find the volume

Write the final answer

The volume of the cube is .

Worked example 6: Finding the volume of a triangular prism

Find the volume of the triangular prism:

Find the area of the base

Multiply the area of the base by the height of the solid to find the volume

Write the final answer

The volume of the triangular prism is .

Worked example 7: Finding the volume of a cylinder

Find the volume of the following cylinder (correct to 1 decimal place):

Find the area of the base

Multiply the area of the base by the height of the solid to find the volume

Write the final answer

The volume of the cylinder is .

Exercise 13.3

Calculate the volumes of the following prisms (correct to decimal place):

The figure here is a triangular prism. The height of the prism is units; the triangles, which both contain right angles, have sides which are , and units long. Calculate the volume of the figure. Round to two decimal places if necessary.

The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Find the volume of the figure.

The picture below shows a cylinder. The height of the cylinder is units; the radius of the cylinder is units. Determine the volume of the figure. Round your answer to two decimal places.

13.3 Right pyramids, right cones and spheres

Pyramid

A pyramid is a geometric solid that has a polygon as its base and faces that converge at a point called the apex. In other words the faces are not perpendicular to the base.

The triangular pyramid and square pyramid take their names from the shape of their base. We call a pyramid a “right pyramid” if the line between the apex and the centre of the base is perpendicular to the base. Cones are similar to pyramids except that their bases are circles instead of polygons. Spheres are solids that are perfectly round and look the same from any direction.

Examples of a square pyramid, a triangular pyramid, a cone and a sphere:

Surface area of pyramids, cones and spheres

Square pyramid

Triangular pyramid

Right cone

Sphere

Worked example 8: Finding the surface area of a triangular pyramid

Find the surface area of the following triangular pyramid (correct to one decimal place):

Find the area of the base

To find the height of the base triangle we use the theorem of Pythagoras:

Find the area of the sides

Find the sum of the areas

Write the final answer

The surface area of the triangular pyramid is .

Worked example 9: Finding the surface area of a cone

Find the surface area of the following cone (correct to 1 decimal place):

Find the area of the base

Find the area of the walls

To find the slant height, , we use the theorem of Pythagoras:

Find the sum of the areas

Write the final answer

The surface area of the cone is .

Worked example 10: Finding the surface area of a sphere

Find the surface area of the following sphere (correct to 1 decimal place):

Worked example 11: Examining the surface area of a cone

If a cone has a height of and a base of radius , show that the surface area is:

Sketch and label the cone

Identify the faces that make up the cone

The cone has two faces: the base and the walls. The base is a circle of radius and the walls can be opened out to a sector of a circle:

This curved surface can be cut into many thin triangles with height close to (where is the slant height). The area of these triangles or sectors can be summed as follows:

Calculate

can be calculated using the theorem of Pythagoras:

Calculate the area of the circular base ()

Calculate the area of the curved walls ()

Find the sum of the areas

Exercise 13.4

Find the total surface area of the following objects (correct to 1 decimal place if necessary):

We first need to find by constructing the vertical (perpendicular) height and using the theorem of Pythagoras:

Now we can find the surface area:

The figure here is a cone. The vertical height of the cone is units and the slant height of the cone is units; the radius of the cone is shown, units. Calculate the surface area of the figure. Round your answer to two decimal places.

Therefore the surface area for the cone is square units.

The figure here is a sphere. The radius of the sphere is units. Calculate the surface area of the figure. Round your answer to two decimal places.

Therefore the surface area is square units.

The figure here shows a pyramid with a square base. The sides of the base are each units long. The vertical height of the pyramid is units, and the slant height of the pyramid is units. Determine the surface area of the pyramid.

The surface area for the pyramid is square units.

Volume of pyramids, cones and spheres

Square pyramid

Triangular pyramid

Right cone

Sphere

This video gives an example of calculating the volume of a sphere.

Video: 2GST

Worked example 12: Finding the volume of a square pyramid

Find the volume of a square pyramid with a height of 3 cm and a side length of 2 cm.

Sketch and label the pyramid

Select the correct formula and substitute the given values

We are given and , therefore

Write the final answer

The volume of the square pyramid is .

Worked example 13: Finding the volume of a triangular pyramid

Find the volume of the following triangular pyramid (correct to 1 decimal place):

Sketch the base triangle and calculate its area

The height of the base triangle () is:

The area of the base triangle is:

Sketch the side triangle and calculate pyramid height

Calculate the volume of the pyramid

Write the final answer

The volume of the triangular pyramid is .

Worked example 14: Finding the volume of a cone

Find the volume of the following cone (correct to decimal place):

Find the area of the base

Calculate the volume

Write the final answer

The volume of the cone is .

Worked example 15: Finding the volume of a sphere

Find the volume of the following sphere (correct to 1 decimal place):

Use the formula to find the volume

Write the final answer

The volume of the sphere is .

Worked example 16: Finding the volume of a complex object

A triangular pyramid is placed on top of a triangular prism, as shown below. The base of the prism is an equilateral triangle of side length and the height of the prism is . The pyramid has a height of . Calculate the total volume of the object.

Calculate the volume of the prism

First find the height of the base triangle using the theorem of Pythagoras:

Next find the area of the base triangle:

Now we can find the volume of the prism:

Calculate the volume of the pyramid

The area of the base triangle is equal to the area of the base of the pyramid.

Calculate the total volume

Therefore the total volume of the object is .

Worked example 17: Finding the surface area of a complex object

With the same complex object as in the previous example, you are given the additional information that the slant height = . Now calculate the total surface area of the object.

Calculate the surface area of each exposed face of the pyramid

Because the base triangle is equilateral, each face has the same base, and therefore the same surface area. Therefore the surface area for each face of the pyramid is .

Calculate the surface area of each side of the prism

Each side of the prism is a rectangle with base and height .

Because the base triangle is equilateral, each side of the prism has the same area. Therefore the surface area for each side of the prism is .

Calculate the total surface area of the object

Therefore the total surface area (of the exposed faces) of the object is .

This video shows an example of calculating the volume of a complex object.

Video: 2GSV

Exercise 13.5

The figure below shows a sphere. The radius of the sphere is units. Determine the volume of the figure. Round your answer to two decimal places.

Therefore the volume for the sphere is units³.

The figure here is a cone. The vertical height of the cone is units and the slant height is units; the radius of the cone is shown, units. Calculate the volume of the figure. Round your answer to two decimal places.

Therefore the volume of the cone is units³.

The figure here is a pyramid with a square base. The vertical height of the pyramid is units and the slant height is units; each side of the base of the pyramid is units. Round your answer to two decimal places.

Therefore the volume of the square pyramid is: units³.

Find the volume of the following objects (round off to 1 decimal place if needed):

We are given the radius of the cone and the slant height. We can use this to find the vertical height () of the cone:

Now we can calculate the volume of the cone:

We first need to find the vertical (perpendicular) height of the triangle () using the theorem of Pythagoras:

Now we can find the volume:

Find the surface area and volume of the cone shown here. Round your answers to the nearest integer.

The surface area of the cone is:

For the volume we first need to find the perpendicular (or vertical) height using the theorem of Pythagoras:

Now we can calculate the volume of the cone:

Therefore the surface area is and the volume is .

Calculate the following properties for the pyramid shown below. Round your answers to two decimal places.

Surface area

We first calculate the vertical (perpendicular) height of the base triangle:

Now we can calculate the surface area of the pyramid:

Therefore the surface area of the triangular pyramid is: .

Volume

We first need to find the vertical height ():

Now we can find the volume:

Therefore the volume of the pyramid is: .

The solid below is made up of a cube and a square pyramid. Find its volume and surface area (correct to decimal place):

The height of the cube is . Since the total height of the object is , the height of the pyramid must be .

We will work out the volume first:

For the surface area we note that one face of the cube is covered by the pyramid. We also note that the base of the pyramid is covered by the cube. So we only need to find the area of 5 sides of the cube and the four triangular faces of the pyramid.

For the triangular faces we need the slant height. We can calculate this using the theorem of Pythagoras:

The surface area is:

The surface area is and the volume is .

13.4 The effect of multiplying a dimension by a factor of k

When one or more of the dimensions of a prism or cylinder is multiplied by a constant, the surface area and volume will change. The new surface area and volume can be calculated by using the formulae from the preceding section.

It is possible to see a relationship between the change in dimensions and the resulting change in surface area and volume. These relationships make it simpler to calculate the new volume or surface area of an object when its dimensions are scaled up or down.

Consider a rectangular prism of dimensions , and . Below we multiply one, two and three of its dimensions by a constant factor of and calculate the new volume and surface area.

Dimensions

Volume

Surface

Original dimensions

Multiply one dimension by

Multiply two dimensions by

Multiply all three dimensions by

Multiply all three dimensions by k

Worked example 18: Calculating the new dimensions of a rectangular prism

Consider a rectangular prism with a height of and base lengths of .

Calculate the surface area and volume.

Calculate the new surface area () and volume () if the base lengths are multiplied by a constant factor of .

Express the new surface area and volume as a factor of the original surface area and volume.

Calculate the original volume and surface area

Calculate the new volume and surface area

Two of the dimensions are multiplied by a factor of 3.

Express the new dimensions as a factor of the original dimensions

Worked example 19: Multiplying the dimensions of a rectangular prism by

Prove that if the height of a rectangular prism with dimensions , and is multiplied by a constant value of , the volume will also increase by a factor .

Calculate the original volume

We are given the original dimensions , and and so the original volume is .

Calculate the new volume

The new dimensions are , , and and so the new volume is:

Write the final answer

If the height of a rectangular prism is multiplied by a constant , then the volume also increases by a factor of .

Worked example 20: Multiplying the dimensions of a cylinder by

Consider a cylinder with a radius of and a height of . Calculate the new volume and surface area (expressed in terms of and ) if the radius is multiplied by a constant factor of .

Calculate the original volume and surface area

Calculate the new volume and surface area

The new dimensions are and .

Exercise 13.6

If the length of the radius of a circle is a third of its original size, what will the area of the new circle be?

The area of the original circle is: . Now we reduce the radius by a third. In other words we multiply by one third. The new area is:

Therefore, if the radius of a circle is a third of its original size, the area of the new circle will be the original area.

If the length of the base's radius and height of a cone is doubled, what will the surface area of the new cone be?

We can find the new area by noting that the area will change by a factor of when we change the dimensions of the cone. In this case we are changing two dimensions of the cone and so the new area will be:

The value of comes from the word “doubled” in the question: the value of is 2.

So the new area of the cone will be if we double the height and the base's radius of the cone.

Therefore the surface area of the new cone will be 4 times the original surface area.

If the height of a prism is doubled, how much will its volume increase by?

We do not know if we have a rectangular prism or a triangular prism. However we do know that the volume of a prism is given by:

Now we are changing just one dimension of the prism: the height. Therefore the new volume is given by:

Therefore the volume of the prism doubles if the height is doubled.

Describe the change in the volume of a rectangular prism if the:

length and breadth increase by a constant factor of .

The volume of a rectangular prism is given by . If we increase the length and breadth by a constant factor of 3 the volume is:

Therefore the volume of the prism increases by a factor of 9 when the length and breadth are increased by a constant factor of 3.

length, breadth and height are multiplied by a constant factor of .

The volume of a rectangular prism is given by . If we increase the length, breadth and height by a constant factor of 3 the volume is:

Therefore the volume of the prism increases by a factor of 27 when the length, breadth and height are increased by a constant factor of 3.

If the length of each side of a triangular prism is quadrupled, what will the volume of the new triangular prism be?

When multiplied by a factor of the volume of a shape will increase by . We are told that the dimensions are quadrupled. This means that each dimension is multiplied by 4. Therefore .

Now we can calculate .

Therefore, if each side of a triangular prism is quadrupled, the volume of the new triangular prism will be 64 times the original shape's volume.

Given a prism with a volume of and a surface area of , find the new surface area and volume for a prism if all dimensions are increased by a constant factor of .

We are increasing all the dimensions by 4 and so the volume will increase by . The surface area will increase by .

Therefore the volume is and the surface area is .

13.5 Chapter summary

Presentation: 2GTH

Area is the two dimensional space inside the boundary of a flat object. It is measured in square units.

Area formulae:

square:

rectangle:

triangle:

trapezium:

parallelogram:

circle:

A right prism is a geometric solid that has a polygon as its base and vertical sides perpendicular to the base. The base and top surface are the same shape and size. It is called a “right” prism because the angles between the base and sides are right angles.

A triangular prism has a triangle as its base, a rectangular prism has a rectangle as its base, and a cube is a rectangular prism with all its sides of equal length. A cylinder is another type of right prism which has a circle as its base.

Surface area is the total area of the exposed or outer surfaces of a prism.

A net is the unfolded “plan” of a solid.

Volume is the three dimensional space occupied by an object, or the contents of an object. It is measured in cubic units.

Volume formulae for prisms and cylinders:

Volume of a rectangular prism:

Volume of a triangular prism:

Volume of a square prism or cube:

Volume of a cylinder:

A pyramid is a geometric solid that has a polygon as its base and sides that converge at a point called the apex. The sides are not perpendicular to the base.

The triangular pyramid and square pyramid take their names from the shape of their base. We call a pyramid a “right pyramid” if the line between the apex and the centre of the base is perpendicular to the base. Cones are similar to pyramids except that their bases are circles instead of polygons. Spheres are solids that are perfectly round and look the same from any direction.

Surface area formulae for right pyramids, right cones and spheres:

square pyramid:

triangular pyramid:

right cone:

sphere:

Volume formulae for right pyramids, right cones and spheres:

square pyramid:

triangular pyramid:

right cone:

sphere:

Multiplying one or more dimensions of a prism or cylinder by a constant affects the surface area and volume.

Exercise 13.7

Find the area of each of the shapes shown. Round your answer to two decimal places if necessary.

We first need to find the height:

Now we can find the area of the parallelogram. Note that the length of the base is .

Find an expression for the area of this figure in terms of . The dimensions of the figure are labelled and . Write your answer in expanded form (not factorised).

Find an expression for the area of this figure in terms of . The figure has dimensions of and , as labelled. Write your answer in expanded form (not factorised).

The figure here is a triangular prism. The height of the prism is units; the triangles, which are both right triangles, have sides which are , and units long. Find the surface area of the figure.

A triangular prism is made up of 2 triangles and 3 rectangles. In this case the triangles are right-angled triangles and so we have the height of the triangle. We also note that each rectangle has a different length and breadth.

The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Find the surface area of the figure.

A rectangular prism is made up of 6 rectangles. In this case there are 4 rectangles with a breadth of 5 units and a height of 8 units and two rectangles with a breadth of 5 units and a height of 5 units.

A cylinder is shown below. The height of the cylinder is ; the radius of the cylinder is , as shown. Find the surface area of the figure. Round your answer to two decimal places.

A cylinder is made up of two circles and a rectangle. We can find the area of each of these and add them up to find the surface area of the cylinder. For the rectangle we note that the length is the circumference of the circle.

The figure here is a triangular prism. The height of the prism is units; the triangles, which both contain right angles, have sides which are , and units long. Determine the volume of the figure.

The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Calculate the volume of the figure.

The picture below shows a cylinder. The height of the cylinder is ; the radius of the cylinder is . Calculate the volume of the figure. Round your answer two decimal places.

The figure here is a sphere. The radius of the sphere is units. Find the surface area of the figure. Round your answer two decimal places.

The figure here shows a pyramid with a square base. The sides of the base are each units long. The vertical height of the pyramid is units, and the slant height of the pyramid is units. Determine the surface area of the pyramid.

The total surface area for the pyramid is: square units.

The figure here is a cone. The vertical height of the cone is units and the slant height of the cone is units; the radius of the cone is shown, units. Find the surface area of the figure. Round your answer two decimal places.

Therefore the total surface area for the cone is square units.

The figure below shows a sphere. The radius of the sphere is units. Determine the volume of the figure. Round your answer to two decimal places.

Therefore the volume for the sphere is units³.

The figure here is a cone. The vertical height of the cone is units and the slant height is units; the radius of the cone is shown, units. Find the volume of the figure. Round your answer to two decimal places.

Therefore the volume is units³.

The figure here is a pyramid with a square base. The vertical height of the pyramid is units and the slant height is units; each side of the base of the pyramid is units. Find the volume of the figure. Round your answer to two decimal places.

Therefore the volume is: units cubed

Consider the solids below:

Calculate the surface area of each solid.

Cone

We first need to calculate the slant height:

Now we can calculate the surface area:

Square pyramid

We first need to calculate the slant height:

Now we can calculate the surface area:

Half sphere

For a half sphere we need to divide the surface area of a sphere by 2. We also need to include the area of a circle.

The surface area of each of the objects is: .

Calculate the volume of each solid.

Cone

Square pyramid:

Half sphere

The volume of a half sphere is half the volume of a sphere.

The volume of each of the objects is:

If the length of each side of a square is a quarter of its original size, what will the area of the new square be?

When we multiply the sides of a square by a factor of the area of the square will increase by .

In this case we are making each side of the square a quarter of the original size so we get:

Therefore, if each side of a square is a quarter of its original size, the area of the new square will be times the original square's area.

If the length of each side of a square pyramid is a third of its original size, what will the surface area of the new square pyramid be?

When we multiply two dimensions of a square pyramid by a factor of the area of the square pyramid will change by .

In this case the length each side of the square pyramid is a third of the original size so we get:

Therefore, if each side of a square pyramid is a third of its original size, the surface area of the new square pyramid will be times the original shape's surface area.

If the length of the base's radius and the height of a cylinder is halved, what will the volume of the new cylinder be?

In this case the base's radius and the height of a cylinder is half of the original size so we get:

Therefore, if the base's radius and the height of a cylinder is halved, the volume of the new cylinder will be times the original shape's volume.

Consider the solids below and answer the questions that follow (correct to decimal place, if necessary):

Calculate the surface area of each solid.

Cylinder

A cylinder is composed of two circles and a rectangle. The breadth of the rectangle is the circumference of the circle.

Triangular prism

A triangular prism is composed of three rectangles and two triangles. We are given the vertical height of the triangles as well as the slant height.

Rectangular prism

A rectangular prism is composed of 6 rectangles. We have the dimensions of all the rectangles.

The surface area of each shape is: .

Calculate volume of each solid.

Cylinder

Triangular prism

Rectangular prism

The volume of each shape is: .

If each dimension of the solids is increased by a factor of , calculate the new surface area of each solid.

Cylinder

Triangular prism

Rectangular prism

The new surface area of each shape is: .

If each dimension of the solids is increased by a factor of , calculate the new volume of each solid.

Cylinder

Triangular prism

Rectangular prism

The new volume of each shape is:

The solid below is made of a cube and a square pyramid. Answer the following:

Find the surface area of the solid shown. Give your answers to two decimal places.

Start with the faces of the cube, which are all squares:

Next we note that the height of the pyramid is:

And we need to calculate the slant height using the theorem of Pythagoras:

Now we can calculate the area of each of the four triangles:

Finally we can calculate the total surface area:

Therefore the surface area is: .

Now determine the volume of the shape. Give your answer to the nearest integer value.

Volume of the pyramid:

Volume of the cube:

Total volume:

Therefore the total volume is: .

Calculate the volume and surface area of the solid below (correct to 1 decimal place):

Surface area

Cylinder:

Cone:

Total surface area: .

Volume

Cylinder:

Cone:

Total volume .

The total surface area and volume is and respectively.

Find the volume and surface areas of the following composite shapes.

The shape is a half sphere on top of a right cone. We can calculate the volume of a cone and add this to half the volume of a sphere. The volume is:

For the surface area we first need to find the slant height:

We have a half sphere on top of a cone. The half sphere covers the circle on top of the cone and so we need to exclude this part from our calculation. For the half sphere we can use half the surface area of a sphere as this does not include the circle at the base of the half sphere.

The surface area is:

Therefore the volume and surface area are and respectively.

We have a cylinder with two half spheres. We can calculate the volume of a cylinder and add the volume of a sphere to this. The volume is:

For the surface area of the two half spheres we can use the surface area of a sphere. For the cylinder we need to exclude the area of the two circles from our calculation since these are covered up by the two half spheres. The surface area is:

Therefore the volume and surface area are: and respectively.

This shape consists of a triangular prism and a rectangular prism. The volume is:

For the surface area we need to exclude the base of the triangular prism as well as part of the top of the rectangular prism.

We first need to calculate the slant height for the triangular prism:

Now we can calculate the surface area of the triangular prism. Remember that we do not need to include the base in our calculation so we only have 2 triangles and 2 rectangles.

For the rectangular prism we can calculate the full surface area and then subtract the base of triangular prism from this.

Now we can add the two surface areas together to get the total surface area:

The volume and surface area are and respectively.

An ice-cream cone (right cone) has a radius of and a height of . A half scoop of ice-cream (hemisphere) is placed on top of the cone. If the ice-cream melts, will it fit into the cone? Show all your working.

We can draw a quick sketch of the problem:

Now we can calculate the volume of the cone and the volume of the ice-cream. The scoop of ice-cream is a half sphere and so the volume of this is half the volume of a sphere.

Yes, the ice-cream will fit into the cone if it melts since the volume of the ice-cream is less than the volume of the cone.

A receptacle filled with petrol has the shape of an inverted right circular cone of height and base radius of . A certain amount of fuel is siphoned out of the receptacle leaving a depth of cm.

Show that .

We can draw the following two triangles based on the information in the figure:

These two triangles are similar triangles. They are both right-angled and share a common angle.

Therefore we can use the ratio of the sides to find :

Determine the volume of fuel that has been siphoned out. Express your answer in litres if

The volume of fuel that has been siphoned out is the total volume of fuel minus the volume of fuel left. The volume of a cone is . From the previous question we have the vertical height for both cones.

Find the volume and surface area of the following prisms.

We are given the diameter of the cylinder. The radius is half the diameter.

Therefore the volume and surface area are and respectively.

This is a triangular pyramid. We are given the vertical height as well as an angle. Since it is a right-angled triangle we can use trigonometry to help us find the missing length.

We redraw the triangle we are interested in:

Now we can calculate (the slant height) and (the base):

Now we know all the lengths we need to know to calculate the volume.

And the surface area is:

The volume and surface area are: and respectively.

Let: , , , , and .

We can view this shape as three rectangular prisms. Two of the three prisms are exactly the same. The volume is therefore:

For the surface area we have several different rectangles. Each of the smaller prisms has 5 exposed rectangles. The larger rectangular prism has 4 rectangles that are not covered up by the smaller prisms. The remaining two rectangles are partly covered up by the smaller prisms and so can be considered as 4 separate rectangles.

We will start by finding the surface area of one of the smaller prisms:

For the larger prism we get:

Therefore the total surface area is:

The volume and surface area are and 886 respectively.

Determine the volume of the following:

We first need to find the vertical height ():

is a square, , .

We first find the vertical height:

We also need to find the length of the side of the square. To do this we note that triangle is a right-angled isosceles triangle. So we can find the length of the side of the square using the theorem of Pythagoras:

Now we can find the volume:

The prism alongside has the following dimensions:

units, units, units. is an arc of a circle with centre . .

Explain why , the radius of the arc , is units.

Since is the centre of the circle (they are both radii of the arc).

and joins and , therefore .

We also know that and since , . Therefore is 4 units.

Calculate the area of the shaded surface.

We have just calculated that . We also know that and so is a square (). This means that we have the area of a square plus one quarter the area of a circle.

The total area is:

Find the volume of the prism.

The area of the shaded piece is the area of the base. For the volume we know that we can calculate the volume by multiplying the area of the base and the height.

You can also calculate the volume using the volume of a rectangular prism and one quarter of the volume of a cylinder.

A cooldrink container is made in the shape of a pyramid with an isosceles triangular base. This is known as a tetrahedron. The angle of elevation of the top of the container is . ; .

Show that the length is cm.

Find the height (to the nearest unit).

Calculate the area of .

Hint: construct a perpendicular line from U to

Find the volume of the container

is a right-angled triangle. We can use trigonometry to help us find . In this case we will use the cosine ratio as we have the hypotenuse () and are looking for the adjacent side ().

is a right-angled triangle. We can use trigonometry to help us find . In this case we will use the sine ratio as we have the hypotenuse () and are looking for the opposite side ().

We first find :

Now we can find the area:

The container is filled with the juice such that an % gap of air is left. Determine the volume of the juice.

To find the volume of the juice we need to multiply the total volume of the container by the percentage of juice in the container.

Below is a diagram of The Great Pyramid.

This is a square-based pyramid and is the centre of the square.

and . The length of the side of the pyramid and the height of the pyramid is .

Determine the area of the base of the pyramid in terms of .

Calculate to decimal places.

From your calculation in question (b) determine .

Determine the volume and surface area of the pyramid.

The volume and surface area are: and respectively.

Chapter 14: Probability

This chapter covers the use of probability models to compare relative frequency to theoretical probability. Venn diagrams are introduced and used to answer probability problems. Intersection, union, mutually exclusive events and complementary events are all introduced.

The difference between theoretical probability and relative frequency should be carefully explained.

The terminology and usage of language in this section can be confusing, especially to second-language speakers. Discuss terminology regularly and emphasise the careful reading of questions.

Union and intersection symbols have been included, but “and” and “or” is the preferred notation in CAPS.

Make sure to outline the differences between “and”, “or”, “only” and “both”. For example, there may be no difference between tea and coffee drinkers and tea or coffee drinkers in common speech but in probability, the “and” and “or” have very specific meanings. Tea and coffee drinkers refers to the intersection of tea drinkers with coffee drinkers, i.e. those who drink both beverages, while tea or coffee drinkers refers to the union, i.e. those who drink only tea, those who drink only coffee and those who drink both.

We use probability to describe uncertain events. When you accidentally drop a slice of bread, you don't know if it's going to fall with the buttered side facing upwards or downwards. When your favourite sports team plays a game, you don't know whether they will win or not. When the weatherman says that there is a % chance of rain tomorrow, you may or may not end up getting wet. Uncertainty presents itself to some degree in every event that occurs around us and in every decision that we make.

We will see in this chapter that all of these uncertainties can be described using the rules of probability theory and that we can make definite conclusions about uncertain processes.

Figure 14.1: Tracking a superstorm. Meteorologists use computer software to help them track storms and predict the weather.

We'll use three examples of uncertain processes to help you understand the meanings of the different words used in probability theory: tossing a coin, rolling dice, and a soccer match.

The following video introduces the concepts used in probability.

Video: 2GVW

Experiment

An experiment refers to an uncertain process.

Outcome

An outcome of an experiment is a single result of that experiment.

Experiment 1: A coin is tossed and it lands with either heads (H) or tails (T) facing upwards. An example outcome of tossing a coin is that it lands with heads facing up:

Experiment 2: Two dice are rolled and the total number of dots added up. An example outcome of rolling two dice:

Experiment 3: Two teams play a soccer match and we are interested in the final score. An example outcome of a soccer match:

Sample space

The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space is denoted with the symbol and the size of the sample space (the total number of possible outcomes) is denoted with

Even though we are usually interested in the outcome of an experiment, we also need to know what the other outcomes could have been. Let's have a look at the sample spaces of each of our three experiments.

Experiment 1: Since a coin can land in one of only two ways (we will ignore the possibility that the coin lands on its edge), the sample space is the set . The size of the sample space of the coin toss is :

Experiment 2: Each of the dice can land on a number from to . In this experiment the sample space of all possible outcomes is every possible combination of the numbers on the first die with the numbers on the second die. This gives a total of possible outcomes. The figure below shows all of the outcomes in the sample space of rolling two dice:

Experiment 3: Each soccer team can get an integer score from upwards. Usually we don't expect a score to go much higher than goals, but there is no reason why this cannot happen. So the sample space of this experiment consists of all possible combinations of two non-negative integers. The figure below shows all of the possibilities. Since we do not limit the score of a team, this sample space is infinitely large:

When we represent a sample space containing real numbers we can either write out all the outcomes in the sample space: or we can represent the sample space as: .

Event

An event is a specific set of outcomes of an experiment that you are interested in. An event is denoted with the letter and the number of outcomes in the event with .

Experiment 1: Let us say that we would like the coin to land heads up. Here the event contains a single outcome: . The size of the event set is .

Experiment 2: Let us say that we are interested in the sum of the dice being . In this case the event set is:

Experiment 3: We would like to know whether the first team will win. For this event to happen the first score must be greater than the second.

. This event set is infinitely large.

14.1 Theoretical probability

Probability

A probability is a real number between and that describes how likely it is that an event will occur.

We can describe probabilities in three ways:

As a real number between 0 and 1. For example .

As a percentage. For example can be written as %.

As a fraction. For example can also be written as .

We note the following about probabilities:

A probability of means that an event will never occur.

A probability of means that an event will always occur.

A probability of means that an event will occur half the time, or time out of every .

When all of the possible outcomes of an experiment have an equal chance of occurring, we can compute the exact theoretical probability of an event. The probability of an event is the ratio between the number of outcomes in the event set and the number of possible outcomes in the sample space.

The following video shows an example of calculating the theoretical probabilities of an event.

Video: 2GVX

Worked example 1: Theoretical probabilities

What is the theoretical probability of each of the events in the first two of our three experiments?

Write down the value of

Experiment 1 (coin):

Experiment 2 (dice):

Write down the size of the event set

Experiment 1:

Experiment 2:

Compute the theoretical probability

Experiment 1:

Experiment 2:

Note that we do not consider the theoretical probability of the third experiment. The third experiment is different from the first two in an important way, namely that all possible outcomes (all final scores) are not equally likely. For example, we know that a soccer score of 1–1 is quite common, while a score of 11–15 is very, very rare. Because all outcomes are not equally likely, we cannot use the ratio between and to compute the theoretical probability of a team winning.

Exercise 14.1

A learner wants to understand the term “event”. So the learner rolls 2 dice hoping to get a total of 8. Which of the following is the most appropriate example of the term “event”?

event set

event set

event set

We recall the definition of the term “event”:

An event is a specific set of outcomes of an experiment that you are interested in.

Therefore the most appropriate example of the term “event” is event set .

A learner wants to understand the term “sample space”. So the learner rolls a die. Which of the following is the most appropriate example of the term “sample space”?

We recall the definition of the term “sample space”:

The sample space of an experiment is the set of all possible outcomes of all experiment.

Therefore the most appropriate example of the term “sample space” is

A learner finds a 6 sided die and then rolls the die once on a table. What is the probability that the die lands on either 1 or 2?

Write your answer as a simplified fraction.

Finally, we can find the probability:

Therefore, the probability that the die lands on either 1 or 2 .

A learner finds a textbook that has 100 pages. He then selects one page from the textbook. What is the probability that the page has an odd page number?

Write your answer as a decimal (correct to 2 decimal places).

Finally, we can find the probability:

Therefore, the probability that the page has an odd page number .

Even numbers from to are written on cards. What is the probability of selecting a multiple of , if a card is drawn at random?

There are 50 cards. They are all even.

All even numbers that are also multiples of 5 are multiples of 10: .

There are of them.

Therefore, the probability of selecting a card that is a multiple of 5 is .

A bag contains red balls, blue balls, green balls and white ball. A ball is picked at random. Determine the probability that it is:

red

blue or white

not green

not green or red

A playing card is selected randomly from a pack of cards. Determine the probability that it is:

the of hearts

a red card

Half the deck is red and half the deck is black.

a picture card

There are picture cards in a suit and suits.

an ace

There are aces in a pack.

a number less than

For each suit of cards there are cards less than : A, and .

14.2 Relative frequency

Relative frequency

The relative frequency of an event is defined as the number of times that the event occurs during experimental trials, divided by the total number of trials conducted.

The relative frequency is not a theoretical quantity, but an experimental one. We have to repeat an experiment a number of times and count how many times the outcome of the trial is in the event set. Because it is experimental, it is possible to get a different relative frequency every time that we repeat an experiment.

The following video explains the concept of relative frequency using the throw of a dice.

Video: 2GW7

Worked example 2: Relative frequency and theoretical probability

We toss a coin times and observe the outcomes. The results of the trials are shown in the table below.

trial

outcome

H

T

T

T

H

T

H

H

H

T

trial

outcome

H

T

T

H

T

T

T

H

T

T

trial

outcome

H

H

H

T

H

T

H

T

T

T

What is the relative frequency of observing heads after each trial and how does it compare to the theoretical probability of observing heads?

Count the number of positive outcomes

A positive outcome is when the outcome is in our event set. The table below shows a running count (after each trial ) of the number of positive outcomes we have observed. For example, after trials we have observed heads times and tails times and so the positive outcome count is .

Compute the relative frequency

Since the relative frequency is defined as the ratio between the number of positive trials and the total number of trials,

The relative frequency of observing heads, , after having completed coin tosses is:

From the last entry in this table we can now easily read the relative frequency after trials, namely . The relative frequency is close to the theoretical probability of . In general, the relative frequency of an event tends to get closer to the theoretical probability of the event as we perform more trials.

A much better way to summarise the table of relative frequencies is in a graph:

The graph above is the plot of the relative frequency of observing heads, , after having completed coin tosses. It was generated from the table of numbers above by plotting the number of trials that have been completed, , on the -axis and the relative frequency, , on the -axis. In the beginning (after a small number of trials) the relative frequency fluctuates a lot around the theoretical probability at , which is shown with a dashed line. As the number of trials increases, the relative frequency fluctuates less and gets closer to the theoretical probability.

Worked example 3: Relative frequency and theoretical probability

While watching soccer games where Team 1 plays against Team 2, we record the following final scores:

Trial

Team 1

Team 2

What is the relative frequency of Team 1 winning?

In this experiment, each trial takes the form of Team 1 playing a soccer match against Team 2.

Count the number of positive outcomes

We are interested in the event where Team 1 wins. From the table above we see that this happens times.

Compute the relative frequency

The total number of trials is . This means that the relative frequency of the event is

It is important to understand the difference between the theoretical probability of an event and the observed relative frequency of the event in experimental trials. The theoretical probability is a number that we can compute if we have enough information about the experiment. If each possible outcome in the sample space is equally likely, we can count the number of outcomes in the event set and the number of outcomes in the sample space to compute the theoretical probability.

The relative frequency depends on the sequence of outcomes that we observe while doing a statistical experiment. The relative frequency can be different every time we redo the experiment. The more trials we run during an experiment, the closer the observed relative frequency of an event will get to the theoretical probability of the event.

So why do we need statistical experiments if we have theoretical probabilities? In some cases, like our soccer experiment, it is difficult or impossible to compute the theoretical probability of an event. Since we do not know exactly how likely it is that one soccer team will score goals against another, we can never compute the theoretical probability of events in soccer. In such cases we can still use the relative frequency to estimate the theoretical probability, by running experiments and counting the number of positive outcomes.

You can use this Phet simulation on probability to do some experiments with dropping a ball through a triangular grid.

Exercise 14.2

A die is tossed 44 times and lands 5 times on the number 3.

What is the relative frequency of observing the die land on the number 3? Write your answer correct to 2 decimal places.

Recall the formula:

Identify variables needed:

Calculate the relative frequency:

Therefore, the relative frequency of observing the die on the number 3 is .

A coin is tossed 30 times and lands 17 times on heads.

What is the relative frequency of observing the coin land on heads? Write your answer correct to 2 decimal places.

Recall the formula:

Identify variables needed:

Calculate the relative frequency:

Therefore, the relative frequency of observing the coin on heads is .

A die is tossed 27 times and lands 6 times on the number 6.

What is the relative frequency of observing the die land on the number 6? Write your answer correct to 2 decimal places.

Recall the formula:

Identify variables needed:

Calculate the relative frequency:

Therefore, the relative frequency of observing the die on the number 6 is .

14.3 Venn diagrams

A Venn diagram is a graphical way of representing the relationships between sets. In each Venn diagram a set is represented by a closed curve. The region inside the curve represents the elements that belong to the set, while the region outside the curve represents the elements that are excluded from the set.

Venn diagrams are helpful for thinking about probability since we deal with different sets. Consider two events, and , in a sample space . The diagram below shows the possible ways in which the event sets can overlap, represented using Venn diagrams:

The sets are represented using a rectangle for and circles for each of and . In the first diagram the two events overlap partially. In the second diagram the two events do not overlap at all. In the third diagram one event is fully contained in the other. Note that events will always appear inside the sample space since the sample space contains all possible outcomes of the experiment.

This video shows how to draw a Venn diagram using a deck of cards as the sample space.

Video: 2GWC

You can use an online tool such as this one to generate Venn diagrams.

Worked example 4: Venn diagrams

Represent the sample space of two rolled dice and the following two events using a Venn diagram:

Event A: the sum of the dice equals

Event B: at least one of the dice shows

Worked example 5: Venn diagrams

Consider the set of diamonds removed from a deck of cards. A random card is selected from the set of diamonds.

Write down the sample space, , for the experiment.

What is the value of ?

Consider the following two events:

: An even diamond is chosen

: A royal diamond is chosen

Represent the sample space and events and using a Venn diagram.

Write down the sample space S

Write down the value of

Draw the Venn diagram

Exercise 14.3

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the event set of . They get stuck, and you offer to help them find it.

Which of the following sets best describes the event set of ?

The event set can be shaded as follows:

Therefore the event set best describes the event set of .

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the event set of . They get stuck, and you offer to help them find it.

Which of the following sets best describes the event set of ?

The event set can be shaded as follows:

Therefore the event set best describes the event set of .

Pieces of paper labelled with the numbers to are placed in a box and the box is shaken. One piece of paper is taken out and then replaced.

What is the sample space, ?

or .

Write down the set , representing the event of taking a piece of paper labelled with a factor of .

.

Write down the set , representing the event of taking a piece of paper labelled with a prime number.

.

Represent , and by means of a Venn diagram.

Find:

Let denote the set of whole numbers from to , denote the set of even numbers from to and denote the set of prime numbers from to . Draw a Venn diagram depicting , and .

There are Grade learners at school. All of these take some combination of Maths, Geography and History. The number who take Geography is , those who take History is , and take Maths. The number who take Maths and History is ; the number who take Geography and History is , and there are who take Maths only and who take History only.

Draw a Venn diagram to illustrate all this information.

We are told that learners take Maths and History. Out of these 16 learners some take Geography as well and some do not.

We are also told that learners take Geography and History. Out of these 6 learners some take Maths as well and some do not.

Let the number of learners who take Maths, History and Geography . Then we can draw the Venn diagram as follows:

How many learners take Maths and Geography but not History?

In the above Venn diagram the number of learners who take Maths and Geography but not History is indicated by . To find we first need to determine .

To find we note that the total number of learners who take History is equal to the sum of each of the following:

The number of learners who take History only: 16

The number of learners who take History and Maths but not Geography:

The number of learners who take History and Geography but not Maths:

The number of learners who take all three subjects:

Now we can find using the same method as to find . This time we will use the total number of learners who take Maths.

Therefore learners take Maths and Geography but not History.

How many learners take Geography only?

Now we need to find . We will use the total number of learners who take Geography to find .

Therefore learners take Geography only.

How many learners take all three subjects?

When we drew the Venn diagram we let be the number of learners that take all three subjects. We calculated in the first question. Therefore learners take all three subjects.

14.4 Union and intersection

Union

The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as or “”.

Intersection

The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as or “”.

The figure below shows the union and intersection for different configurations of two events in a sample space, using Venn diagrams.

The unions and intersections of different events. Note that in the middle column the intersection, , is empty since the two sets do not overlap. In the final column the union, , is equal to and the intersection, , is equal to since is fully contained in .

Exercise 14.4

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the event set of the intersection between event set and event set , also written as . They get stuck, and you offer to help them find it.

Which set best describes the event set of ?

The intersection between event set and event set , also written as , can be shaded as follows:

Therefore the event set best describes the event set of .

A group of learners are given the following Venn diagram:

The sample space can be described as

They are asked to identify the event set of the union between event set and event set , also written as . They get stuck, and you offer to help them find it.

Which set best describes the event set of ?

The union between event set and event set , also written as , can be shaded as follows:

Therefore the event set best describes the event set of .

14.5 Probability identities

By definition, the sample space contains all possible outcomes of an experiment. So we know that the probability of observing an outcome from the sample space is .

We can calculate the probability of the union of two events using:

We will prove this identity using the Venn diagrams given above.

For each of the terms in the union and intersection identity, we can draw the Venn diagram and then add and subtract the different diagrams. The area of a region represents its probability.

We will do this for the first column of the Venn diagram figure given previously. You should also try it for the other columns.

This video gives an example of how we can add probabilities together.

Video: 2GWN

Worked example 6: Union and intersection of events

Relate the probabilities of events and from Example 4 (two rolled dice) and show that they satisfy the identity

Write down the probabilities of the two events, their union and their intersection

From the Venn diagram in Example 4, we can count the number of outcomes in each event. To get the probability of an event, we divide the size of the event by the size of the sample space, which is .

Write down and check the identity

Exercise 14.5

A group of learners is given the following event sets:

The sample space can be described as .

They are asked to calculate the value of . They get stuck, and you offer to calculate it for them. Give your answer as a decimal number, rounded to two decimal places.

Identify variables needed:

Calculate :

Therefore, the value of is .

A group of learners is given the following event sets:

The sample space can be described as .

They are asked to calculate the value of . They get stuck, and you offer to calculate it for them. Give your answer as a decimal number, rounded to two decimal value.

Make the subject, and we get:

Identify variables needed:

Calculate :

Therefore, the value of is .

14.6 Mutually exclusive events

Mutually exclusive events

Two events are called mutually exclusive if they cannot occur at the same time. Whenever an outcome of an experiment is in the first event it cannot also be in the second event, and vice versa.

Another way of saying this is that the two event sets, and , cannot have any elements in common, or (where denotes the empty set). We have already seen the Venn diagrams of mutually exclusive events in the middle column of the Venn diagrams provided earlier.

From this figure you can see that the intersection has no elements. You can also see that the probability of the union is the sum of the probabilities of the events.

This relationship is true for mutually exclusive events only.

Worked example 7: Mutually exclusive events

We roll two dice and are interested in the following two events:

The sum of the dice equals

At least one of the dice shows

Show that the events are mutually exclusive.

Draw the sample space and the two events

Determine the intersection

From the above figure we notice that there are no elements in common in A and B. Therefore the events are mutually exclusive.

Exercise 14.6

State whether the following events are mutually exclusive or not.

A fridge contains orange juice, apple juice and grape juice. A cooldrink is chosen at random from the fridge. Event A: the cooldrink is orange juice. Event B: the cooldrink is apple juice.

We are choosing just one cooldrink from the fridge. This cooldrink cannot be both an orange juice and an apple juice. Therefore these two events are mutually exclusive.

A packet of cupcakes contains chocolate cupcakes, vanilla cupcakes and red velvet cupcakes. A cupcake is chosen at random from the packet. Event A: the cupcake is red velvet. Event B: the cupcake is vanilla.

We are choosing just one cupcake from the packet. This cupcake cannot be both a red velvet cupcake and a vanilla one. Therefore these two events are mutually exclusive.

A card is chosen at random from a deck of cards. Event A: the card is a red card. Event B: the card is a picture card.

We are choosing just one card from the deck. This card can be both a red card and a picture card. Therefore these two events are not mutually exclusive.

A cricket team plays a game. Event A: they win the game. Event B: they lose the game.

The cricket team can either win the game or lose the game. They cannot simultaneously win and lose the game. Therefore these two events are mutually exclusive.

Note that a tie game does not count as either a win or a loss. In a tie neither team can be said to have won the match.

14.7 Complementary events

Complementary set

The complement of a set, , is a new set that contains all of the elements that are not in . We write the complement of as , or sometimes .

For an experiment with sample space and an event we can derive some identities for complementary events. Since every element in is not in , we know that complementary events are mutually exclusive.

Since every element in the sample space is either in or in , the union of complementary events covers the sample space.

From the previous two identities, we also know that the probabilities of complementary events sum to .

Worked example 8: Reasoning with Venn diagrams

In a survey people were questioned about which product they use: A or B or both. The report of the survey shows that people use product A, people use product B and people use neither. Use a Venn diagram to work out how many people:

use product A only

use product B only

use both product A and product B

Summarise the sizes of the sample space, the event sets, their union and their intersection

We are told that people were questioned, so the size of the sample space is .

We are told that people use product A, so .

We are told that people use product B, so .

We are told that people use neither product. This means that people use at least one of the two products, so .

We are not told how many people use both products, so we have to work out the size of the intersection, , by using the identity for the union of two events:

Determine whether the events are mutually exclusive

Since the intersection of the events, , is not empty, the events are not mutually exclusive. This means that their circles should overlap in the Venn diagram.

Draw the Venn diagram and fill in the numbers

Read off the answers

people use product A only.

people use product B only.

people use both products.

Exercise 14.7

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the complementary event set of , also known as . They get stuck, and you offer to help them find it.

Which of the following sets best describes the event set of ?

The event set can be shaded as follows:

The complementary event set can be shaded as follows:

Therefore the event set best describes the complementary event set of , also known as .

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the complementary event set of , also known as . They get stuck, and you offer to help them find it.

Which of the following sets best describes the event set of ?

The event set can be shaded as follows:

The complementary event set can be shaded as follows:

Therefore the event set best describes the complementary event set of , also known as .

Given the following Venn diagram:

The sample space can be described as .

Are and mutually exclusive?

We recall the definition of the term “mutually exclusive”:

Two events are called mutually exclusive if they cannot occur at the same time.

The event set for is:

The event set for is:

The question we must ask: Can they occur at the same time?

By observing both sets, we can identify the following overlapping event set: .

Therefore, yes, the event sets and are mutually exclusive in this example.

Given the following Venn diagram:

The sample space can be described as .

Are and mutually exclusive?

We recall the definition of the term “mutually exclusive”:

Two events are called mutually exclusive if they cannot occur at the same time.

The event set for is:

The event set for is:

The question we must ask: Can they occur at the same time?

By observing both sets, we can identify the following overlapping event set:

Therefore, no, the event sets and are not mutually exclusive in this example.

14.8 Chapter summary

This video summarises the concepts covered in this chapter. Note that some of the examples used in this video may not be suited to all learners.

Presentation: 2GX2

An experiment refers to an uncertain process.

An outcome of an experiment is a single result of that experiment.

The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space is denoted with the symbol and the size of the sample space (the total number of possible outcomes) is denoted with .

An event is a specific set of outcomes of an experiment that you are interested in. An event is denoted with the letter and the number of outcomes in the event with .

A probability is a real number between and that describes how likely it is that an event will occur.

A probability of means that an event will never occur.

A probability of means that an event will always occur.

A probability of means that an event will occur half the time, or time out of every .

A probability can also be written as a percentage or as a fraction.

When all of the possible outcomes of an experiment have an equal chance of occurring, we can compute the exact theoretical probability of an event. The probability of an event is the ratio between the number of outcomes in the event set and the number of possible outcomes in the sample space.

The relative frequency of an event is defined as the number of times that the event occurs during experimental trials, divided by the total number of trials conducted.

The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as or .

The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as or .

The probability of observing an outcome from the sample space is 1: .

The probability of the union of two events is calculated using: .

Mutually exclusive events are two events that cannot occur at the same time. Whenever an outcome of an experiment is in the first event, it can not also be in the second event.

The complement of a set, , is a different set that contains all of the elements that are not in . We write the complement of as or “”.

Complementary events are mutually exclusive: .

Complementary events cover the sample space:

Probabilities of complementary events sum to : .

Exercise 14.8

A learner wants to understand the term “outcome”. So the learner rolls a die. Which of the following is the most appropriate example of the term “outcome”?

A teacher walks into the class room.

The die lands on the number 5.

The clock strikes 3 pm.

We recall the definition of the term “outcome”:

An outcome of an experiment is a single result of that experiment

Therefore the most appropriate example of the term “outcome” is: the die lands on the number 5.

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the event set of . They get stuck, and you offer to help them find it.

Which of the following sets best describes the event set of ?

The event set can be shaded as follows:

Therefore the event set best describes the event set of .

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the event set of the union between event set and event set , also written as . They get stuck, and you offer to help them find it.

Write down the event set that best describes .

The union between event set and event set , also written as , can be shaded as follows:

Therefore the event set best describes the event set of .

Given the following Venn diagram:

The sample space can be described as .

Are and mutually exclusive?

We recall the definition of the term “mutually exclusive”:

Two events are called mutually exclusive if they cannot occur at the same time.

The event set for is: .

The event set for is: .

The question we must ask: Can they occur at the same time?

By observing both sets, we can identify the following overlapping event set:

Therefore, yes, the event sets and are mutually exclusive in this example.

A group of learners are given the following Venn diagram:

The sample space can be described as .

They are asked to identify the complementary event set of , also known as . They get stuck, and you offer to help them find it.

Write down the set that best describes the event set of .

The event set can be shaded as follows:

The complementary event set can be shaded as follows:

Therefore the event set best describes the complementary event set of , also known as .

A learner finds a deck of 52 cards and then takes one card from the deck. What is the probability that the card is a king?

Write your answer as a decimal (correct to 2 decimal places).

Finally, we calculate the probability:

Therefore, the probability that the card is a King .

A die is tossed 21 times and lands 2 times on the number 3.

What is the relative frequency of observing the die land on the number 3? Write your answer correct to 2 decimal places.

Recall the formula:

Identify variables needed:

Calculate the relative frequency:

Therefore, the relative frequency of observing the die on the number 3 is .

A coin is tossed 44 times and lands 22 times on heads.

What is the relative frequency of observing the coin land on heads? Write your answer correct to 2 decimal places.

Recall the formula:

Identify variables needed:

Calculate the relative frequency:

Therefore, the relative frequency of observing the coin on heads is .

A group of children were asked if they eat Frosties, Strawberry Pops or both. children said they eat both and said they only eat Frosties. What is the probability that a child chosen at random will eat only Strawberry Pops?

In a group of learners, all but had a packet of chips or a cooldrink or both. If had a packet of chips and of these also had a cooldrink, what is the probability that one learner chosen at random has:

both chips and cooldrink

only cooldrink

Since learners had at least one, and learners had a packet of chips, then learners only had a cooldrink.

A box contains coloured blocks. The number of each colour is given in the following table.

Colour

Purple

Orange

White

Pink

Number of blocks

24

32

41

19

A block is selected randomly. What is the probability that the block will be:

purple

Before we answer the questions we first work out how many blocks there are in total. This gives us the sample space: .

The probability that a block is purple is:

purple or white

The probability that a block is either purple or white is:

pink and orange

Since one block cannot be two colours the probability of this event is .

not orange

We first work out the probability that a block is orange:

The probability that a block is not orange is:

A small nursery school has a class with children of various ages. The table gives the number of children of each age in the class.

3 years old

4 years old

5 years old

Male

Female

If a child is selected at random what is the probability that the child will be:

a female

We calculate the total number of pupils at the school: .

The total number of female children is .

The probability of a randomly selected child being female is:

a 4 year old male

The probability of a randomly selected child being a 4 year old male is:

aged or

There are children aged or . The probability of a randomly selected child being either or is: .

aged and

A child cannot be both and , so the probability is .

not

This is the same as a randomly selected child being either or and so is .

either or female

The probability of a child being either or female is:

Fiona has labelled discs, which are numbered from to . If a disc is selected at random what is the probability that the disc number:

ends with

The set of all discs ending with is: . This has elements.

The probability of drawing a disc that ends with is:

is a multiple of

The set of all discs that are multiples of is: . This has elements.

The probability of drawing a disc that is a multiple of is:.

is a multiple of

The set of all discs that are multiples of is: . This set has elements.

The probability of drawing a disc that is a multiple of is: .

is number

There is only one element in this set and so the probability of drawing is: .

is not a multiple of

The set of all discs that are a multiple of is: . This set contains elements. Therefore the number of discs that are not multiples of is: .

The probability of drawing a disc that is not a multiple of is: .

is a multiple of or

In part b), we worked out the probability for a disc that is a multiple of . Now we work out the number of elements in the set of all discs that are multiples of : . This has elements.

The probability that a disc is a multiple of either or is:

is a multiple of and

The set of all discs that are multiples of and is the same as the set of all discs that are a multiple of . Therefore the probability of drawing a disc that is both a multiple of and is: .

is number

There is only element in this set and so the probability is .

Use a Venn diagram to work out the following probabilities for a die being rolled:

a multiple of and an odd number

For a die being rolled the sample space is

There is only one possibility here: the die lands on a 5. Therefore the probability is: .

a number that is neither a multiple of nor an odd number

There is only one multiple of 5 in the sample space, this is an odd number. Therefore the set of numbers that are neither a multiple of 5 nor an odd number is: . Therefore the probability is: .

a number which is not a multiple of , but is odd

There is only one multiple of 5 in the sample space, this is an odd number. Therefore the set of numbers that are not a multiple of 5 but is an odd number is: . Therefore the probability is: .

A packet has yellow sweets and pink sweets. The probability of taking out a pink sweet is . What is the probability of taking out a yellow sweet?

In a car park with cars, there are Opels. What is the probability that the first car to leave the car park is:

an Opel

not an Opel

Nezi has loose socks in a drawer. Eight of these are plain orange and two are plain pink. The remaining socks are neither orange nor pink. Calculate the probability that the first sock taken out at random is:

orange

not orange

pink

not pink

orange or pink

neither orange nor pink

A plate contains shortbread cookies, ginger biscuits, chocolate chip cookies and Jambos. If a biscuit is selected at random, what is the probability that:

it is either a ginger biscuit or a Jambo

Total number of biscuits is .

it is not a shortbread cookie

tickets were sold at a raffle. Jabulile bought tickets. What is the probability that Jabulile:

wins the prize

does not win the prize

A group of children were surveyed to see how many had red hair and brown eyes. children had red hair but not brown eyes, children had brown eyes and red hair, children had brown eyes but not red hair and children did not have brown eyes or red hair.

How many children were in the school?

The following possibilities exist for the hair and eye colour surveyed. Each of these is mutually exclusive:

A child has brown eyes and red hair.

A child has brown eyes but not red hair.

A child has red hair but not brown eyes.

A child does not have red hair or brown eyes.

Since we are given the total number of children in each of these four groups we can add these together to get the total number of children: .

What is the probability that a child chosen at random has:

brown eyes

red hair

A child with brown eyes is chosen randomly. What is the probability that this child will have red hair?

A jar has purple sweets, blue sweets and green sweets in it. The probability that a sweet chosen at random will be purple is and the probability that it will be green is .

If I choose a sweet at random what is the probability that it will be:

purple or blue

green

purple

Same as not green:

If there are sweets in the jar how many purple ones are there?

of the purple sweets in (b) have streaks on them and the rest do not. How many purple sweets have streaks?

Box A contains cards numbered , and .

Box B contains cards numbered and .

One card is removed at random from each box.

Find the probability that:

the sum of the numbers is .

the sum of the two numbers is a prime number.

the product of the two numbers is at least .

the sum is equal to the product.

A card is drawn at random from an ordinary pack of playing cards.

Find the probability that the card drawn is:

the three of diamonds

the three of diamonds or any heart

a diamond or a three

The card drawn is the three of diamonds. It is placed on the table and a second card is drawn. What is the probability that the second card drawn is not a diamond.

A group of learners is given the following event sets:

The sample space can be described as

They are asked to calculate the value of . They get stuck, and you offer to calculate it for them. Give your answer as a decimal number, rounded to two decimal value.

Make the subject, and we get:

Identify variables needed:

Calculate :

Therefore, the value of is .

For each of the following, draw a Venn diagram to represent the situation and find an example to illustrate the situation.

a sample space in which there are two events that are not mutually exclusive

An example is drawing a card from deck of cards. We can draw a red card that is also a picture card.

a sample space in which there are two events that are complementary

An example is rolling a die. The event of rolling an even number is complementary to the event of rolling an odd number.

Use a Venn diagram to prove that the probability of either event A or B occurring (A and B are not mutually exclusive) is given by:

All the clubs are taken out of a pack of cards. The remaining cards are then shuffled and one card chosen. After being chosen, the card is replaced before the next card is chosen.

What is the sample space?

Find a set to represent the event, , of drawing a picture card.

Find a set for the event, , of drawing a numbered card.

Represent the above events in a Venn diagram.

What description of the sets and is suitable? (Hint: Find any elements of in and of in .)

Mutually exclusive and complementary.

A survey was conducted at Mutende Primary School to establish how many of the learners buy vetkoek and how many buy sweets during break. The following was found:

learners bought nothing

learners bought vetkoek

learners bought sweets

Represent this information with a Venn diagram.

The following Venn diagram represents the given information. However we can calculate more information from this that will help us answer the second part of this question.

We note the following information:

400 learners bought vetkoek, some of these also bought sweets.

300 learners bought sweets, some of these also bought vetkoek.

Of the total number of learners, 50 did not buy anything, so bought either vetkoek or sweets or both.

Let the number of learners who bought vetkoek only be , the number of learners who bought sweets only be and the number of learners who bought both be . Now we note the following:

We can fill this in on the Venn diagram:

If a learner is chosen randomly, calculate the probability that this learner buys:

sweets only

vetkoek only

neither vetkoek nor sweets

vetkoek and sweets

vetkoek or sweets

In a survey at Lwandani Secondary School, people were questioned to find out how many read the Sowetan, how many read the Daily Sun and how many read both. The survey revealed that read the Daily Sun, read the Sowetan and read neither. Use a Venn diagram to find the percentage of people that read:

only the Daily Sun

The following Venn diagram represents the given information. However we can calculate more information from this that will help us answer the problem.

We note the following information:

45 people read the Daily Sun, some of these also read the Sowetan.

30 people read the Sowetan, some of these also read the Daily Sun.

Of the total number of people questioned 10 did not read either newspaper, so read neither.

Let the number of people who read the Daily Sun only be , the number of people who read the Sowetan only be and the number of people who read both be . Now we note the following:

We can fill this in on the Venn diagram:

.

only the Sowetan

.

both the Daily Sun and the Sowetan

.

In a class there are

8 learners who play football and hockey

7 learners who do not play football or hockey

13 learners who play hockey

19 learners who play football

How many learners are there in the class?

Let and be the learners who play hockey and football respectively.

Of people, have an interest in reading magazines and have an interest in reading books, have an interest in reading both magazines and books.

Represent the information in a Venn diagram.

Let and be the people who read books and magazines respectively.

How many people have no interest in reading magazines or books?

11 people only read magazines, 6 people only read books and 6 people read both.

The total number of people is 36. Therefore we can find the number of people who have no interest in reading magazines or books:

If a person is chosen at random from the group, find the probability that the person will:

have an interest in reading magazines and books.

have an interest in reading books only.

not have any interest in reading books.

learners were surveyed and the following information was revealed from this group:

learners take Geography

learners take French

learners take History, but take neither Geography nor French.

In addition the following Venn Diagram has been filled in below:

Let be the event that a learner takes Geography.

Let be the event that a learner takes French.

Let be the event that a learner takes History.

From the information above, determine the values of , , and .

Therefore , , and .

Determine the probability that a learner chosen at random from this group:

takes only Geography.

takes French and History, but not Geography.

List of image attributions

Untitled by Sandstein licenced under CC-BY 3.0 unported licence at http://en.wikipedia.org/wiki/File:SAM_PC_1_-_Tally_sticks_1_-_Overview.jpg

Square kilometre array by UCL mathematics and physical science licenced under CC-BY 2.0 generic licence at https://www.flickr.com/photos/uclmaps/13221058125/

Indian blanket sunflower by Audrey licenced under CC-BY 2.0 licence at https://www.flickr.com/photos/audreyjm529/206614588/

First use of an equals sign ever licenced under Public domain licence on http://en.wikipedia.org/wiki/File:First_Equation_Ever.png

Untitled by NASA licenced under Public Domain licence at http://en.wikipedia.org/wiki/File:GPS_Satellite_NASA_art-iif.jpg

AJ LBW by Barry Gill licenced under CC-BY 2.0 generic unported licence at https://www.flickr.com/photos/42482371@N00/7529096134/in/photolist-6Gue7W-ctjzYj-bFL19k-xS21X-6SLCuz-6SLD4x-6SQFgC-6SLHRp-6945Yu

P. Oxy. I 29 by anonymous licenced under Public Domain licence at http://en.wikipedia.org/wiki/File:P._Oxy._I_29.jpg

Screenshot from http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html licenced under Public Domain

JSE by Alastair Hay licenced under CC-BY licence

Screenshot of Siyavula page stats by Heather Williams licenced under Public Domain licence

Clinometerlow by Kamal Child licenced under Public Domain at http://en.wikipedia.org/wiki/File:Clinometerlow.jpg

LapanganTenis Pusdiklat by Ahmed Fauzi licenced under Public Domain licence at http://en.wikipedia.org/wiki/File:LapanganTenis_Pusdiklat.jpg

Tracking a Superstorm by NASA Goddard Space Flight Centre licenced under CC-BY 2.0 generic licence at https://www.flickr.com/photos/gsfc/8971451260

Final score by apasciuto licenced under CC-BY 2.0 generic licence at https://www.flickr.com/photos/apasciuto/5996076302/in/photolist-8nVmhf-8nTuoh-a8RszU-9xwnFE-9xwc4W-6sD2WZ-6NbUAG