Year 10 Interactive Maths - Second Edition

## Similar Figures

Similar figures have the same shape (but not necessarily the same size) and the following properties:
• Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal.
• Corresponding angles are equal.

For example, consider the following squares.

Thus the squares are similar figures as their corresponding sides are proportional and their corresponding angles are equal.

###### Note:
• Each side of figure PQRS has been multiplied by 2 to obtain the sides of figure ABCD. The number 2 is called the scale factor.
• Similar figures are equiangular (i.e. the corresponding angles of similar figures are equal).

## Similar Triangles

Similar triangles can be applied to solve real world problems. For example, similar triangles can be used to find the height of a building, the width of a river, the height of a tree etc.

###### Recall that:

If two triangles are similar, then:

• they are equiangular
• the corresponding sides are in the same ratio
• the angle included between any two sides of one triangle is equal to the angle included between the corresponding sides of the other triangle

#### Example 10

Find the value of x in the following pair of triangles.

##### Solution:

###### Note:

Corresponding angles are marked in the same way in diagrams.

#### Example 11

Find the value of the pronumeral in the following diagram.

### Problem Solving

#### Example 12

Find the value of the height, h m, in the following diagram at which the tennis ball must be hit so that it will just pass over the net and land 6 metres away from the base of the net.

##### Solution:

So, the height at which the ball should be hit is 2.7 m.

#### Example 13

Adam looks in a mirror and sees the top of a building. His eyes are 1.25 m above ground level, as shown in the following diagram.

If Adam is 1.5 m from the mirror and 181.5 m from the base of the building, how high is the building?

##### Solution:

So, the height of the building is 150 m.

###### Note:

a. Equal angles are marked in the same way in diagrams.

b. Two triangles are similar if:

• two pairs of corresponding sides are in the same ratio and the angle included between the sides is the same for both triangles.
• the corresponding sides are in the same ratio.
• the corresponding angles are the same.