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# Enlargement and reduction transformations

## 24.7 Enlargement and reduction transformations

We can use transformations to change the size of a shape. To make a shape bigger, we use an enlargement transformation. This is sometimes referred to as stretching or expanding a shape. To make a shape smaller, we use a reduction transformation, also referred to as shrinking a shape.

### Part A: Finding the image of a rectangle under an enlargement transformation

The diagram shows rectangle $$ABCD$$ and its image $$A'B'C'D'$$ under an enlargement transformation with the origin as the centre of enlargement.

1. Compare the sides of rectangle $$ABCD$$ and the corresponding sides of its image $$A'B'C'D'$$:

1. $\dfrac{A'B'}{AB} = \dots$
2. $\dfrac{B'C'}{BC} = \dots$
3. $\dfrac{C'D'}{CD} = \dots$
4. $\dfrac{D'A'}{DA} = \dots$
2. What do you notice about the ratios?
3. Compare the vertices of rectangle $$ABCD$$ and the corresponding vertices of its image $$A'B'C'D'$$.

1. $$A = (\dots; \dots)$$ and $$A' = (\dots; \dots)$$
2. $$B = (\dots; \dots)$$ and $$B' = (\dots; \dots)$$
3. $$C = (\dots; \dots)$$ and $$C' = (\dots; \dots)$$
4. $$D = (\dots; \dots)$$ and $$D' = (\dots; \dots)$$
4. What do you notice about the values of the corresponding vertices?
5. Calculate the perimeter of rectangle $$ABCD$$ and the perimeter of rectangle $$A'B'C'D'$$. What can you deduce?
6. Complete the sentence: Perimeter of rectangle $$ABCD = \dots \times$$ perimeter of rectangle $$A'B'C'D'$$.
7. Calculate the area of rectangle $$ABCD$$ and the area of rectangle $$A'B'C'D'$$. What can you deduce?
8. Complete the sentence: Area of rectangle $$ABCD = \square^\square \times$$ area of rectangle (\ A'B'C'D' \).
9. How would you describe these two shapes? Are they congruent shapes? Are they similar shapes? Write a sentence to explain your answer.

### Part B: Finding the image of a triangle under a reduction transformation

The diagram shows $$\triangle PQR$$ and its image $$\triangle P'Q'R'$$ under a reduction transformation with the origin as the centre of reduction.

1. Compare the sides of $$\triangle PQR$$ and the corresponding sides of its image $$\triangle P'Q'R'$$:

1. $\dfrac{P'Q'}{PQ} = \dots$
2. $\dfrac{Q'R'}{QR} = \dots$
3. $\dfrac{R'P'}{RP} = \dots$ (Hint: Use the Theorem of Pythagoras)
2. What do you notice about the ratios?
3. Compare the vertices of $$\triangle PQR$$ and the corresponding vertices of its image $$\triangle P'Q'R'$$.

1. $$P = (\dots; \dots)$$ and $$P' = (\dots; \dots)$$
2. $$Q = (\dots; \dots)$$ and $$Q' = (\dots; \dots)$$
3. $$R = (\dots; \dots)$$ and $$R' = (\dots; \dots)$$
4. What do you notice about the values of the corresponding vertices?
5. Calculate the perimeter of $$\triangle PQR$$ and the perimeter of $$\triangle P'Q'R'$$. What can you deduce?
6. Complete the sentence: Perimeter of $$\triangle P'Q'R' = \dots \times$$ perimeter of $$\triangle PQR$$.
7. Calculate the area of $$\triangle PQR$$ and the area of $$\triangle P'Q'R'$$. What can you deduce?
8. Complete the sentence: Area of $$\triangle P'Q'R' = \left(\frac{\square}{\square} \right)^\square \times$$ area of $$\triangle PQR$$.
9. How would you describe these two shapes? Are they congruent shapes? Are they similar shapes? Write a sentence to explain your answer.

### Properties of enlargement and reduction transformations

We have discovered the following properties of enlargement and reduction transformations:

• All the sides of the image are scaled in the same proportion. This proportion is called the scale factor of the transformation.
• For enlargement and reduction transformations we must specify the centre of enlargement or reduction.
• For enlargement transformations, the scale factor is $$> 1$$.
• For reduction transformations, the scale factor is a fraction between $$0$$ and $$1$$.
• When the centre of enlargement or reduction is the origin, the coordinates of the vertices of the image are scaled by the scale factor.
• Perimeter of the image = scale factor $$\times$$ perimeter of the shape.
• Area of the image = (scale factor)$$^2 \times$$ area of the shape
• For enlargement and reduction transformations, the shape and its image are similar shapes. Their shape and orientation are the same, but they differ in size.

## Worked example 24.6: Determining the scale factor for an enlargement transformation

Determine the scale factor for the enlargement transformation with the origin as the centre of enlargement.

The diagram shows $$\triangle WXY$$ and $$\triangle W'X'Y'$$. We can draw dotted lines between the corresponding vertices to help us compare the corresponding sides.

### Compare the lengths of the corresponding sides.

We can compare any pair of corresponding sides, so we choose sides for which we can easily find the lengths. $$XY$$ and $$X'Y'$$ are horizontal lengths and we can find the lengths by subtracting the values of the corresponding $$x$$-coordinates.

\begin{align*} \frac{X'Y'}{XY} &= \frac{15 - 9}{5 - 3} \\ &= \frac{6}{2} \\ &= 3 \end{align*}

We can calculate the lengths of the vertical sides $$XW$$ and $$X'W'$$ by subtracting the values of the corresponding $$y$$-coordinates.

$\frac{W'X'}{WX} = \frac{10 - 4}{4 - 2} = \frac{6}{2} = 3$

We can find the length of $$WY$$ and $$W'Y'$$ using the Theorem of Pythagoras:

\begin{align*} WY^2 &= WX^2 + XY^2 \\ &= (2)^2 + (2)^2 \\ &= 4 + 4 \\ &= 8 \\ WY &= \sqrt{8} \\ &= 2\sqrt{2} \end{align*} \begin{align*} W'Y'^2 &= W'X'^2 + X'Y'^2 \\ &= (6)^2 + (6)^2 \\ &= 36 + 36 \\ &= 72 \\ W'Y' &= \sqrt{72} \\ &= 6\sqrt{2} \end{align*}

So, $\dfrac{W'Y'}{WY} = \dfrac{6\sqrt{2}}{2\sqrt{2}} = 3$

Notice that the lengths of all three sides have been increased in the same proportion.

The scale factor for this enlargement transformation is 3.

# Test yourself now

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Exercise 24.1

For each diagram, describe the enlargement or reduction transformation. Remember to state the centre of enlargement or reduction in each case.

The diagram shows $$\triangle PQR$$ and its image $$\triangle P'Q'R'$$ under an enlargement transformation with the origin as the centre of enlargement.

The diagram shows square $$BCDE$$ and its image $$B'C'D'E'$$ under a reduction transformation with the origin as the centre of reduction.

The diagram shows rectangle $$DEFG$$ and its image $$D'E'F'G'$$ under an enlargement transformation with the origin as the centre of enlargement.

The diagram shows $$\triangle PQR$$ and its image $$\triangle P'Q'R'$$ under a reduction transformation with the origin as the centre of reduction.

The diagram shows square $$QTSR$$ and its image $$Q'T'S'R'$$ under a reduction transformation with the origin as the centre of reduction. Find the scale factor.

Scale factor = $$\frac{1}{2}$$

The diagram shows $$\triangle ABC$$ and its image $$\triangle A'B'C'$$ under an enlargement transformation with the origin as the centre of enlargement. Determine the scale factor.

Scale factor = $$\frac{3}{2}$$

The diagram shows rectangle $$MNPQ$$ and its image $$M'N'P'Q'$$ under an enlargement transformation with the origin as the centre of enlargement. Find the scale factor.

Scale factor = $$\frac{3}{2}$$

The diagram shows square $$WXYZ$$ and its image $$W'X'Y'Z'$$ under an enlargement transformation with the origin as the centre of enlargement. Find the scale factor.

Scale factor = 4