Dilations as Transformations of the Plane

New York State Common Core Math Geometry, Module 2, Lesson 6

Worksheets for Geometry

Student Outcomes

• Students review the properties of basic rigid motions.
• Students understand the properties of dilations and that a dilations is also an example of a transformation of the plane.

Dilations as Transformations of the Plane

Classwork

Exercises 1–6

1. Find the center and the angle of the rotation that takes 𝐴𝐵 to 𝐴′𝐵′. Find the image 𝑃′ of point 𝑃 under this rotation.
2. In the diagram below, △ 𝐵′𝐶′𝐷′ is the image of △ 𝐵𝐶𝐷 after a rotation about a point 𝐴. What are the coordinates of 𝐴, and what is the degree measure of the rotation?
3. Find the line of reflection for the reflection that takes point 𝐴 to point 𝐴′. Find the image 𝑃′ under this reflection.
4. Quinn tells you that the vertices of the image of quadrilateral 𝐶𝐷𝐸𝐹 reflected over the line representing the equation 𝑦 = − 3/2𝑥 + 2 are the following: 𝐶′(−2,3), 𝐷′(0,0), 𝐸′(−3,−3), and 𝐹′(−3,3). Do you agree or disagree with Quinn? Explain.
5. A translation takes 𝐴 to 𝐴′. Find the image 𝑃′ and pre-image 𝑃′′ of point 𝑃 under this translation. Find a vector that describes the translation.
6. The point 𝐶′ is the image of point 𝐶 under a translation of the plane along a vector.
   a. Find the coordinates of 𝐶 if the vector used for the translation is 𝐵𝐴.
   b. Find the coordinates of 𝐶 if the vector used for the translation is 𝐴𝐵.

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Exercises 7 - 9

7. A dilation with center 𝑂 and scale factor 𝑟 takes 𝐴 to 𝐴′ and 𝐵 to 𝐵′. Find the center 𝑂, and estimate the scale factor r.
8. Given a center 𝑂, scale factor 𝑟, and points 𝐴 and 𝐵, find the points 𝐴′ = 𝐷_𝑂,𝑟(𝐴) and 𝐵′ = 𝐷_𝑂,𝑟(𝐵). Compare length 𝐴𝐵 with length 𝐴′𝐵′ by division; in other words, find 𝐴′𝐵′/𝐴𝐵. How does this number compare to 𝑟?
9. Given a center 𝑂, scale factor 𝑟, and points 𝐴, 𝐵, and 𝐶, find the points 𝐴′ = 𝐷𝑂,𝑟(𝐴), 𝐵′ = 𝐷_𝑂,𝑟(𝐵), and 𝐶′ = 𝐷_𝑂,𝑟(𝐶). Compare 𝑚∠𝐴𝐵𝐶 with 𝑚∠𝐴′𝐵′𝐶′. What do you find?

Lesson Summary

• There are two major classes of transformations: those that are distance preserving (translations, reflections, rotations) and those that are not (dilations).
• Like rigid motions, dilations involve a rule assignment for each point in the plane and also have inverse functions that return each dilated point back to itself.

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