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Maths

Congruent and similar shapes

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# Similar shapes

Similar figures are identical in shape, but not in size.

For example, two circles are always similar.

Two squares are always similar:

And two rectangles could be similar:

But will probably not be.

## 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.

Look at the diagram below. Figures abcd and ABCD are similar.

and

Therefore,   ie, the sides are in the same ratio.

We can also say that and

and therefore

These facts can be used when solving problems.

Question

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

Answer

The answer is PS = 15.75cm

Here's how to work it out. We know that the sides are in the same ratio, so:

Include the numbers you already know from the diagram like this:

PS = (7 × 9) ÷ 4 = 63 ÷ 4 = 15.75 cm

Remember : Try to use the formula which has the 'unknown' at the top of the fraction.

Question

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

Answer

The answer is 4.5cm. Here's how to work it out. Because the shapes are similar we can write

xy = (4 × 9) ÷ 8 = 4.5

Question

What is the size of angle ?

Answer

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

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