Topics
9. Dilations & Similarity
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United States of America
Grade 8

9.03 Similarity and transformations

Lesson
Practice

Introduction

We have previously discussed  congruence transformations  . We saw that reflections, rotations, and translations resulted in an image congruent to the original object or shape. Because congruence holds for these transformations, similarity does as well, since all congruent figures can be considered similar with a ratio of 1\text{:}1.

Dilations on the other hand will result in an image which is similar to the original object or shape, but not congruent. Note that not all similar figures are congruent, only those that have a ratio of 1\text{:}1.

Similarity transformations

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

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Triangle A is similar to triangle B.

Triangle B is an enlargement of triangle A.

Examples

Example 1

Consider the figures shown.

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a

Are the two triangles congruent, similar, or neither?

Worked Solution
Create a strategy

Recall that congruent figures have exactly the same shape and size, while similar figures are two figures having the same shape.

Apply the idea

The two triangles have the same shape but not the same size, so the two triangles are similar.

b

What is the transformation from triangle ABC to triangle A'B'C'?

A
Dilation
B
Reflection
C
Rotation
D
Translation
Worked Solution
Create a strategy

Recall the following definitions:

  • A rotation involves turning a figure.

  • A reflection involves flipping a figure.

  • A translation involves sliding a figure.

  • A dilation involves resizing a figure.

Apply the idea

The correct option is A: Dilation, because the triangle ABC is larger than triangle A'B'C' which involves resizing the figure.

c

What is the scale factor of the dilation from triangle ABC to triangle A'B'C'?

Worked Solution
Create a strategy

We need to count the distance of triangles A'B'C' and ABC from the origin. Then we can divide the distance of the larger triangle from the center of dilation by the distance of the smaller triangle from the center of dilation.

Apply the idea
\displaystyle \text{Distance of } A', \,B', \, C' \text{ from origin}\displaystyle =\displaystyle 8Count the distance of the endpoints from (0,0)
\displaystyle \text{Distance of } A, \,B, \,C \text{ from origin}\displaystyle =\displaystyle 4Count the distance of the endpoints from (0,0)
\displaystyle \text{Scale factor}\displaystyle =\displaystyle \dfrac84Divide the distances
\displaystyle =\displaystyle 2Evaluate

Example 2

What is the sequence of transformations from triangle ABC to triangle A''B''C''? Use triangle A'B'C' as a guide.

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A
Translation of 3 units right and 5 units up and rotation of 90 degrees clockwise around the origin
B
Dilation by a factor of 2 with the origin as the center of dilation and rotation of 90 degrees clockwise around the origin
C
Dilation by a factor of 2 with the origin as the center of dilation and translation of 3 units right and 5 units up
D
Reflection about the x-axis and dilation by a factor of 2 with the origin as the center of dilation
Worked Solution
Create a strategy

Recall the following definitions:

  • A rotation involves turning a figure.

  • A reflection involves flipping a figure.

  • A translation involves sliding a figure.

  • A dilation involves resizing a figure.

Apply the idea

The size of the triangles are not the same and A''B''C'' was turned, which means that the transformed triangle has been dilated and rotated.

The answer is option B:

Dilation by a factor of 2 with the origin as the center of dilation and rotation of 90 degrees clockwise around the origin

Idea summary

A two-dimensional figure is similar to another if the second figure can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

Reflections, rotations, and translations resulted in an image congruent to the original image while dilations result in an image which is similar to the original image.

All congruent figures can be considered similar, but not all similar figures are congruent.

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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