Properties of Similarity Transformations

New York State Common Core Math Geometry, Module 2, Lesson 13

Worksheets for Geometry

Student Outcomes

• Students know the properties of a similarity transformation are determined by the transformations that compose the similarity transformation.
• Students are able to apply a similarity transformation to a figure by construction.

Properties of Similarity Transformations

Classwork

Example 1

Similarity transformation 𝐺 consists of a rotation about the point 𝑃 by 90°, followed by a dilation centered at 𝑃 with a scale factor of 𝑟 = 2, and then followed by a reflection across line ℓ. Find the image of the triangle.

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Example 2

A similarity transformation 𝐺 applied to trapezoid 𝐴𝐵𝐶𝐷 consists of a translation by vector 𝑋𝑌, followed by a reflection across line 𝓂, and then followed by a dilation centered at 𝑃 with a scale factor of 𝑟 = 2. Recall that we can describe the same sequence using the following notation: 𝐷𝑃,2 (𝑟𝓂(𝑇𝑋𝑌(𝐴𝐵𝐶𝐷))). Find the image of 𝐴𝐵𝐶𝐷

Exercise 1

A similarity transformation for triangle 𝐷𝐸𝐹 is described by 𝑟𝓃 (𝐷𝐴,1/2(𝑅𝐴,90°(𝐷𝐸𝐹))). Locate and label the image of triangle 𝐷𝐸𝐹 under the similarity.

Lesson Summary

Properties of similarity transformations:

1. Distinct points are mapped to distinct points.
2. Each point 𝑃′ in the plane has a pre-image.
3. There is a scale factor of 𝑟 for 𝐺 so that for any pair of points 𝑃 and 𝑄 with images 𝑃′ = 𝐺(𝑃) and 𝑄′ = 𝐺(𝑄), then 𝑃′𝑄′ = 𝑟𝑃𝑄.
4. A similarity transformation sends lines to lines, rays to rays, line segments to line segments, and parallel lines to parallel lines.
5. A similarity transformation sends angles to angles of equal measure.
6. A similarity transformation maps a circle of radius 𝑅 to a circle of radius 𝑟𝑅, where 𝑟 is the scale factor of the similarity transformation.

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