Home Practice
For learners and parents For teachers and schools
Textbooks
Full catalogue
Leaderboards
Learners Leaderboard Classes/Grades Leaderboard Schools Leaderboard
Pricing Support
Help centre Contact us
Log in

We think you are located in United States. Is this correct?

7.4 Summary

7.4 Summary (EMCHX)

d82d52aeeb2a081c7e4e116972db8ef9.png
Theorem of Pythagoras: \(AB^2 = AC^2 + BC^2\)
Distance formula: \(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Gradient: \(m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{ or } \quad m_{AB} = \frac{y_1 - y_2}{x_1 - x_2}\)
Mid-point of a line segment: \(M(x;y) = \left( \frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2} \right)\)
Points on a straight line: \(m_{AB} = m_{AM} = m_{MB}\)
Straight line equations Formulae
Two-point form: \(\dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Gradient-point form: \(y - y_1 = m (x - x_{1})\)
Gradient-intercept form: \(y = mx + c\)
Horizontal lines: \(y = k\)
Vertical lines \(x = k\)
Parallel lines 9daee11f01ada64c3a1ff39e0a0aa32c.png \(m_1 = m_2\) \(\theta_1 = \theta_2\)
Perpendicular lines d82ed7f3faf04f0ef92da6e1fdae45df.png \(m_1 \times m_2 = -1\) \(\theta_{1} = \text{90} ° + \theta_{2}\)
  • Inclination of a straight line: the gradient of a straight line is equal to the tangent of the angle formed between the line and the positive direction of the \(x\)-axis.

    \[m = \tan \theta \qquad \text{ for } \text{0}° \leq \theta < \text{180}°\]

  • Equation of a circle with centre at the origin:

    If \(P(x;y)\) is a point on a circle with centre \(O(0;0)\) and radius \(r\), then the equation of the circle is:

    \[x^{2} + y^{2} = r^{2}\]
  • General equation of a circle with centre at \((a;b)\):

    If \(P(x;y)\) is a point on a circle with centre \(C(a;b)\) and radius \(r\), then the equation of the circle is:

    \[(x - a)^{2} + (y - b)^{2} = r^{2}\]
  • A tangent is a straight line that touches the circumference of a circle at only one point.

  • The radius of a circle is perpendicular to the tangent at the point of contact.

temp text