Similar Figures

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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.

Square PQRS has sides of 2 cm in length.

Square ABCD has sides of 4 cm in length.

Clearly, AB/PQ = BC/QR = CD/RS = DA/SP = 2

Also, angle A = angle P, angle B = angle Q, angle C = angle R, angle D = angle S.

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.

Right angled triangle DEF has an hypotenuse of length 7.5 cm and another side of length 6 cm.

Right angled triangle ABC has an hypotenuse of unknown length x cm and another side of length 12 cm.
Solution:

Triangle ABC and triangle DEF are similar as they are equiangular. x / 7.5 = 12 / 6 and solving for x gives x = 15.

Note:

Corresponding angles are marked in the same way in diagrams.

Example 11

Find the value of the pronumeral in the following diagram.

Triangle ABC has a height of x cm and a width of 3 cm. Triangle ADE has a height of (x + 4) cm and a width of 6 cm.

Solution:

Triangle ADE and triangle ABC are similar as they are equiangular. Therefore, AD / AB = DE / BC so (x + 4) / x = 6 / 3. Solving for x, we find x = 4.

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.

The net is 0.9 m high. The tennis player strikes the ball at an unknown height, h m, when the ball is 12 m from the net.

Solution:

We draw and label a diagram as shown. Height is measured vertically. So, angle EDA is a right angle. We assume that the net is vertical.

Triangle ABC has a width of 6 m and height of 0.9 m and triangle ADE has a width of (6 m + 12 m) and a height of h m.

Triangle ADE and triangle ABC are similar as they are equiangular. Therefore, h / 0.9 = 18 / 6 and after solving for h, we find h = 2.7.

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.

Adam's eyes are 1.25 m from the ground and he is 1.5 m from the mirror and 181.5 m from the base of the building.

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:

We draw and label the diagram shown below. Height is measured vertically. So, angle MAB and angle MDC are right angles, angle DMC and angle AMB are equal due to the mirror's reflection, and angle DCM and angle ABM are equal by the angle sum of a triangle.

Triangle MDC has a height of 1.25 m and a base of 1.5 m. Triangle MAB has an unknown height of x m and a base of 180 m.

Triangle MAB and triangle MDC are similar as they are equiangular. Therefore, x / 1.25 = 180 / 1.5 and after solving for x, we find x = 150.

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.
Key Terms

similar figures, scale factor, equiangular, similar triangles

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