# What Are Similarity Transformations, and Why Do We Need Them?

### New York State Common Core Math Geometry, Module 2, Lesson 12

Worksheets for Geometry, Module 2, Lesson 12

Student Outcomes

• Students define a similarity transformation as the composition of basic rigid motions and dilations.
• Students define two figures to be similar if there is a similarity transformation that takes one to the other.
• Students can describe a similarity transformation applied to an arbitrary figure (i.e., not just triangles) and can use similarity to distinguish between figures that resemble each other versus those that are actually similar.

What Are Similarity Transformations, and Why Do We Need Them?

Classwork

Opening Exercise

Observe Figures 1 and 2 and the images of the intermediate figures between them. Figures 1 and 2 are called similar. What observations can we make about Figures 1 and 2?

Definition:

A _____ _____ (or ) _____ is a composition of a finite number of dilations or basic rigid motions. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition. If there are no dilations in the composition, the scale factor is defined to be 1.

Definition:

Two figures in a plane are if there exists a similarity transformation taking one figure onto the other figure.

Example 1

Figure 𝑍′ is similar to Figure 𝑍. Describe a transformation that maps Figure 𝑍 onto Figure 𝑍′

Exercises 1–3

1. Figure 1 is similar to Figure 2. Which transformations compose the similarity transformation that maps Figure 1 onto Figure 2?
2. Figure 𝑆 is similar to Figure 𝑆′. Which transformations compose the similarity transformation that maps 𝑆 onto 𝑆′?
3. Figure 1 is similar to Figure 2. Which transformations compose the similarity transformation that maps Figure 1 onto Figure 2?

Example 2

Show that no sequence of basic rigid motions and dilations takes the small figure to the large figure. Take measurements as needed.

Exercises 4–5

1. Is there a sequence of dilations and basic rigid motions that takes the large figure to the small figure? Take measurements as needed.
2. What purpose do transformations serve? Compare and contrast the application of rigid motions to the application of similarity transformations.

Lesson Summary

Two figures are similar if there exists a similarity transformation that maps one figure onto the other. A similarity transformation is a composition of a finite number of dilations or rigid motions.

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