9.01 Identify similar polygons

Consider the triangles below. What do you notice about the sides? What about the angles?

Since all sides in the smaller triangle are the same length, all the angles have the same measure. This is an equilateral triangle.

Now compare the smaller triangle to the larger triangle. What is the same? What is different?

You might have noticed that both triangles have the same angle measures but they have different side lengths. Specifically, each side in the smaller triangle is multiplied by 3 to create the side of the larger triangle. This means that the scale factor is 3. Since all of the sides of the larger triangle are the same length, it is also an equilateral triangle.

All of the corresponding sides between the two shapes are in the same ratio, or proportional, and all corresponding angles between the two shapes are equal. This means the two triangles are similar.

What do you notice about the triangles in the figure below? Can we say that corresponding sides are proportional and corresponding angles are congruent?

Recall that the smaller triangle is equilateral because all of the sides have the same length and all angles have the same measure. To make the larger triangle, two of the sides of the smaller triangle are enlarged by a factor of 3 and one side is enlarged by a factor of 2. This makes the sides of the larger triangle unequal. So the larger triangle is not equilateral, and all angles are not 60\degree.

The corresponding sides between the two shapes are not proportional and all corresponding angles between the two shapes are not equal. This means the triangles are not similar.

Examples

Example 1

Consider the following triangles:

a

Which of these triangles are similar?

A

B

C

D

Worked Solution

Create a strategy

Recall that triangles are similar when either:

• All sides are in the same ratio, or

• All angles are equal, or

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

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

Apply the idea

The corresponding angles of the triangles in options A and C are equal.

The answers are options A and C.

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b

Which of the following conditions satisfied the answer from part (a)?

A

All corresponding angles are equal.

B

Two angles are equal and one side is a multiple of the corresponding side of the other.

C

All corresponding sides are in the same ratio.

Worked Solution

Apply the idea

Answer is option A: All corresponding angles are equal.

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Consider the quadrilaterals below. What do you notice about the sides? What about the angles?

Both shapes are rectangles because they are quadrilaterals with 4 right angles. To make the larger rectangle, two of the sides of the smaller rectangle are enlarged by a factor of 2 and the other two sides are not changed. Are these rectangles similar?

Even though all corresponding angles between the two shapes are equal, the corresponding sides between the two shapes are not proportional. That is, the same scale factor has not been applied to each side of the smaller rectangle to create the lager one. Therefore, the two rectangles are not similar.

Consider the quadrilaterals below. What do you notice about the sides? What about the angles?

To make the larger quadrilateral, each side length of the smaller quadrilateral has been doubled, or multiplied by a factor of 2.

All four corresponding angles are equal, and all corresponding sides are proportional. This means the quadrilaterals are similar.

If the only difference between two shapes is the size, (one is an enlargement of the other) then the two shapes are similar.

Therefore, polygons are similar if:

• All corresponding sides are proportional

• All corresponding angles are equal

Examples

Example 2

Consider the following shapes:

a

Are the two shapes similar?

Worked Solution

Create a strategy

Recall that for the two shapes to be similar, all corresponding angles must be equal, and all sides must be in the same ratio.

Apply the idea

The corresponding angles of the shapes are 79\degree and 78\degree , which are not equal.

So, the two shapes are not similar.

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b

Which of the following conditions satisfied the answer from part (a)?

A

All sides are in the same ratio but not all matching angles are equal.

B

All matching angles are not equal and all sides are not in the same ratio.

C

All matching angles are equal but all sides are not in the same ratio.

Worked Solution

Create a strategy

Use the answer from part (a) and compare the ratios of the corresponding side lengths.

Apply the idea

All four sides of the smaller rhombus have length 20, and all four sides of the larger rhombus have length 81.

\displaystyle \dfrac{81}{20}=\dfrac{81}{20}	\displaystyle =	\displaystyle \dfrac{81}{20}=\dfrac{81}{20}	Compare the ratios of corresponding side lengths
\displaystyle 79\degree	\displaystyle \neq	\displaystyle 78\degree	Compare the corresponding angles

The answer is option A: All sides are in the same ratio but not all matching angles are equal.

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