Properties of Similarity Transformations

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

Worksheets for Geometry, Module 2, Lesson 13

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.

• Show Video for Whole Lesson

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.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.