# Similar figures

Look at these similar figures:

B is an enlargement of A. The lengths have doubled, but the angles have stayed the same.

Remember: For any pair of similar figures, corresponding sides are in the same ratio, and corresponding angles are equal.

## Linear scale factor

The size of an enlargement/reduction is described by its scale factor.

For example, a of 2 means that the new shape is twice the size of the original. A scale factor of 3 means that the new shape is three times the size of the original.

To calculate the Scale Factor, we use the following:

$S{F_{Enlargement}} = \frac{{Big}}{{Small}}$

$S{F_{{\mathop{\rm Re}\nolimits} duction}} = \frac{{Small}}{{Big}}$

You can get the 'big' and 'small' from the corresponding sides on the figures.

### Example

The rectangles pqrs and PQRS are similar. What is the length of PS?

PS is on the bigger rectangle, therefore we will be calculating an enlargement scale factor first.

$S{F_{Enlargement}} = \frac{{Big}}{{Small}} = \frac{7}{4}$

Therefore rectangle PQRS is $$\frac{7}{4}$$ times bigger than rectangle pqrs.

So, $$PS = \frac{7}{4} \times 9 = 15.75cm$$

(You can type in your calculator 7 ÷ 4 × 9)

Question

wxyz and WXYZ are similar figures. What is the length of XY?

XY is on the bigger figure, therefore we will be calculating the enlargement scale factor.

$S{F_{Enlargement}} = \frac{{Big}}{{Small}} = \frac{9}{8}$

Therefore figure WXYZ is $$\frac{9}{8}$$ times bigger than figure wxyz.

So, $$XY = \frac{9}{8} \times 4 = 4.5cm$$

Question

What is the size of angle WXY?

Remember that the angles in similar figures stay the same. So angle WXY is 57°.

Question

The sides of a rectangle measure 8cm and 6cm.

If the rectangle is to be enlarged using scale factor $$\frac{3}{2}$$ what will be the new lengths of the sides?

$\frac{3}{2} \times 8 = 12cm$

$\frac{3}{2} \times 6 = 9cm$

The new lengths will be 12cm and 9cm

Question

The sides of a rectangle measure 20cm and 28cm.

If the rectangle is to be enlarged using scale factor $$\frac{3}{4}$$ what will be the new lengths of the sides?

$\frac{3}{4} \times 20 = 15cm$

$\frac{3}{4} \times 28 = 21cm$

The new lengths will be 15cm and 21cm

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