10.08 Similarity

Shapes or objects are described as similar if they are exactly the same shape but different size. Objects or shapes are congruent if they are exactly the same.

In similar shapes:

• All corresponding sides are proportional

• All corresponding angles are equal

Some of our standard shapes have similarity because of the nature of their properties:

• All circles are similar because every circle is an enlargement of another. That is, one circle can create any other circle by enlarging or reducing its radius.

• All squares are similar because every square is an enlargement of another.

Sometimes a shape can be rotated, translated or reflected but still be similar.

These two following triangles are similar as one is an enlargement of the other. The methods of proving similar triangles are detailed in the section below.

All the corresponding sides between the two shapes are in the same ratio and all corresponding angles between the two shapes are equal.

These triangles are not similar as each side has not been increased by the same factor.

To make the larger triangle, two of the sides of the smaller triangle are enlarged by a factor of 3 and one of the sides is enlarged by a factor of 2. The corresponding sides between the two shapes are not in the same ratio and all corresponding angles between the two shapes are not equal.

These rectangles are not similar for the same reason as the triangles.

The corresponding sides between the two shapes are not in the same ratio. That is, the same enlargement factor has not been applied to each side of the smaller rectangle.

These quadrilaterals are similar.

To make the larger quadrilateral, each side length of the smaller quadrilateral has been doubled. There are four pairs of equal corresponding angles, and all corresponding sides are in the same ratio.

Examples

Example 1

Consider the two similar triangles.

a

State the angles that correspond to \angle D, \, \angle E, and \angle C.

Worked Solution

Create a strategy

Choose the angles that are marked in the same way.

Apply the idea

The following is the complete corresponding angles:

\angle D corresponds to \angle N.

\angle E corresponds to \angle L.

\angle C corresponds to \angle M.

b

Which side does CD corresponds to in \triangle LMN?

A

LN

B

ML

C

MN

Worked Solution

Create a strategy

Look for the side in the other triangle, that is opposite the angle marked in the same way as the angle opposite the given side.

Apply the idea

Corresponding sides appear opposite corresponding angles. The angle that appears opposite side CD is \angle E is \angle L. So in \triangle LMN, we are looking for the side opposite \angle L.

Side CD corresponds to MN, option C.

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c

Which side does CE corresponds to in \triangle LMN?

A

LN

B

MN

C

ML

Worked Solution

Create a strategy

Look for the side in the other triangle, that is opposite the angle marked in the same way as the angle opposite the given side.

Apply the idea

Corresponding sides appear opposite corresponding angles. The angle that appears opposite side CE is \angle D is \angle N. So in \triangle LMN, we are looking for the side opposite \angle N.

Side CE corresponds to ML, option C.

Example 2

Consider the two shapes:

a

Are the two shapes similar?

Worked Solution

Create a strategy

The two shapes are regular hexagons and has the same angles.

Apply the idea

Yes, the two shapes are similar.

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b

Which of the following statements are true?

A

One of the shapes is a rotation of the other.

B

One of the shapes is a translation of the other.

C

One of the shapes is an enlargement of the other.

Worked Solution

Create a strategy

The shapes have the same angles and it only change the size.

Apply the idea

The correct answer is option C: One of the shapes is an enlargement of the other.

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Example 3

The smaller quadrilateral has been reflected, then enlarged and finally rotated.

a

Are the two shapes similar?

Worked Solution

Create a strategy

Remember that reflecting and rotating a shape does not change its angles or side lengths. Enlarging a shape changes the side lengths but keeps all angles the same.

Apply the idea

Yes, the two quadrilaterals are similar.

b

Which side in the larger shape is corresponding to side AB?

A

GH

B

EF

C

FG

D

HE

Worked Solution

Create a strategy

Identify where side AB is in relation to the other sides and angles in the smaller shape, and then look for a side in the larger shape which is in the same corresponding position.

Apply the idea

The side EF is in the same corresponding position as side AB.

So the correct answer is option B.

c

Which angle in the larger shape is corresponding to \angle ADC?

A

\angle EHG

B

\angle FEH

C

\angle HGF

D

\angle GFE

Worked Solution

Create a strategy

Identify where \angle ADC is in relation to the other sides and angles in the smaller shape and then look for a side in the larger shape which is in the same corresponding position.

Apply the idea

The \angle EHG is in the same corresponding position as \angle ADC.

So the correct answer is option A.

In similar triangles, just as any similar object:

• all corresponding angles are equal

• all corresponding sides are in the same ratio

There are three methods for proving two triangles are similar. You only need to use one of these proofs to show two triangles are similar.

AAA (angle, angle, angle):

If all corresponding angles in two different triangles are equal the triangles are similar.

The triangles show that all three corresponding pairs of angles are equal.

\angle A = \angle P,\,\angle B = \angle Q,\,\angle C = \angle R

Therefore, the \triangle ABC is similar to \triangle PQR (AAA). Note that we only need to show two angles are congruent to prove (AAA), as if we have two congruent angles then the third angle must also be congruent as angles in a triangle always add to 180\degree.

SSS (side, side, side):

If two triangles have all three pairs of corresponding sides in the same ratio, then these triangles are similar.

For example, all corresponding sides in the triangle below are in the same ratio:

\dfrac{12}{4}=\dfrac{12}{4}=\dfrac{21}{7}=3

SAS (side, angle, side):

If two triangles have two pairs of sides in the same ratio and equal included angles, then these triangles are similar.

There are two pairs of sides in the same ratio and the included angle (marked by the dot) is equal.

\dfrac{PQ}{AB}=\dfrac{6}{3}=2,\,\angle Q=\angle B,\,\dfrac{QR}{BC}=\dfrac{10}{5}=2

Examples

Example 4

Consider the shapes attached:

a

Are the two shapes similar?

Worked Solution

Create a strategy

Check if all pairs of the corresponding sides have the same ratio and the angles have been preserve.

Apply the idea

Yes, the two shapes are similar.

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b

Find the enlargement factor.

A

6

B

5

C

7

Worked Solution

Create a strategy

Divide the side length of the larger shape by the corresponding side length of the smaller shape.

Apply the idea

\displaystyle \text{Enlargement factor}	\displaystyle =	\displaystyle 18\div 3	Divide the larger length by the smaller length
	\displaystyle =	\displaystyle 6	Evaluate

\displaystyle \text{Enlargement factor}	\displaystyle =	\displaystyle 42\div 7	Divide the larger length by the smaller length
	\displaystyle =	\displaystyle 6	Evaluate

\displaystyle \text{Enlargement factor}	\displaystyle =	\displaystyle 48\div 8	Divide the larger length by the smaller length
	\displaystyle =	\displaystyle 6	Evaluate

The enlargement factor of the two triangles is 6.

So the correct answer is option A.

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Example 5

Which two of these triangles are similar?

A

B

C

D

Worked Solution

Create a strategy

Look for the two triangles that have:

• All corresponding sides in the same ratio, OR

• All corresponding angles are equal, OR

• Two corresponding sides are in the same ratio and the included angles equal, OR

• Right-angles with hypotenuse and one side in the same ratio.

Apply the idea

The triangles in options A and C are similar, because all of the corresponding angles are equal.

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