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 πΊ consists of a rotation about the point π by 90Β°, followed by a dilation centered at π with a scale factor of π = 2, and then followed by a reflection across line β. Find the image of the triangle.

Example 2

A similarity transformation πΊ applied to trapezoid π΄π΅πΆπ· consists of a translation by vector ππ, followed by a reflection across line π, and then followed by a dilation centered at π with a scale factor of π = 2. Recall that we can describe the same sequence using the following notation: π·π,2 (ππ(πππ(π΄π΅πΆπ·))). Find the image of π΄π΅πΆπ·

Exercise 1

A similarity transformation for triangle π·πΈπΉ is described by ππ (π·π΄,1/2(ππ΄,90Β°(π·πΈπΉ))). Locate and label the image of triangle π·πΈπΉ under the similarity.

Lesson Summary

Properties of similarity transformations:

1. Distinct points are mapped to distinct points.
2. Each point πβ² in the plane has a pre-image.
3. There is a scale factor of π for πΊ so that for any pair of points π and π with images πβ² = πΊ(π) and πβ² = πΊ(π), then πβ²πβ² = πππ.
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 π to a circle of radius ππ, where π is the scale factor of the similarity transformation.

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