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4. Dilations & Similarity
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Grade 8

4.02 Identifying similar polygons

Lesson
Practice
Lesson

Recall that a scale drawing is a diagram of an object in which the dimensions are in proportion to the actual dimensions of the object. Shapes can be scaled as well. When one shape is created by scaling another, we call them similar.

Similar Triangles

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$3 to create the side of the larger triangle. This means that the scale factor is $3$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.

 

Non-Similar Triangles

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$3 and one side is enlarged by a factor of $2$2. This makes the sides of the larger triangle unequal. So the larger triangle is not equilateral, and all angles are not $60^\circ$60°.

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.

 

Non-Similar Quadrilaterals

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$4 right angles. To make the larger rectangle, two of the sides of the smaller rectangle are enlarged by a factor of $2$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.

 

Similar Quadrilaterals

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$2.

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

Summary

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

 

Exploration

Explore this interactive to create similar polygons by sliding the blue slider.  

Created with Geogebra

 

Practice questions

Question 1

Consider the following shapes:

Two parallelograms are depicted side by side. The one on the left has an angle at its bottom left corner labeled as 78 degrees and the length of its top side is 15 units. All sides are marked with single hashmarks. The one on the right has an angle at its bottom left corner labeled as 71 degrees and the length of its top side is 31 units. All sides are marked with double hashmarks.

  1. Are the two shapes similar?

    Yes

    A

    No

    B

  2. Give a reason for your answer.

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

    A

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

    B

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

    C

Question 2

Consider the following triangles:

  1. Which of these triangles are similar?

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 53 degrees at the bottom left vertex with a double arc, and 72 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 53-degree angles is labeled with a length of 4 units.

    A

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 70 degrees at the bottom left vertex with a double arc, and 55 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 55-degree angles is labeled with a length of 8 units.

    B

    A triangle is depicted with its interior angles labeled at each vertex: 72 degrees at the top vertex with a single arc, 38 degrees at the bottom left vertex with a double arc, and 70 degrees at the bottom right vertex with a triple arc. The side between the 72-degree and 70-degree angles is labeled with a length of 4 units.

    C

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 70 degrees at the bottom left vertex with a double arc, and 55 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 55-degree angles is labeled with a length of 32 units.

    D

  2. Give a suitable reason for their similarity.

    All corresponding angles are equal.

    A

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

    B

    All corresponding sides are in the same ratio.

    C

 

Outcomes

8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

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