Similar figures

Look at these similar figures:

Diagram of 2 shapes of the same proportions but different sizes

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 scale factor 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?

Two rectangles, one measuring 4cm x 9cm, the other with a height of 7cm and unknown length

Answer

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?

Two similar shapes, one 8cm x 4cm with an x angle of 57°, the other with a height of 9cm

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?

Two similar shapes, one 8cm x 4cm with an x angle of 57°, the other with a height of 9cm

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?

Rectangle 8 x 6cm

\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?

Rectangle 28 x 20cm

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

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

The new lengths will be 15cm and 21cm

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