New York State Common Core Math Geometry, Module 2, Lesson 6
Worksheets for Geometry, Module 2, Lesson 6
- 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
- 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.
- 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?
- Find the line of reflection for the reflection that takes point 𝐴 to point 𝐴′. Find the image 𝑃′ under this reflection.
- 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.
- 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.
- 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.
Exercises 7 - 9
- 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.
- Given a center 𝑂, scale factor 𝑟, and points 𝐴 and 𝐵, find the points 𝐴′ = 𝐷𝑂,𝑟(𝐴) and 𝐵′ = 𝐷𝑂,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?
- Given a center 𝑂, scale factor 𝑟, and points 𝐴, 𝐵, and 𝐶, find the points 𝐴′ = 𝐷𝑂,𝑟(𝐴), 𝐵′ = 𝐷𝑂,𝑟(𝐵), and 𝐶′ = 𝐷𝑂,𝑟(𝐶). Compare 𝑚∠𝐴𝐵𝐶 with 𝑚∠𝐴′𝐵′𝐶′. What do you find?
- 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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