# Dilations as Transformations of the Plane

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

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 1β6

1. Find the center and the angle of the rotation that takes π΄π΅ to π΄β²π΅β². Find the image πβ² of point π under this rotation.
2. In the diagram below, β³ π΅β²πΆβ²π·β² is the image of β³ π΅πΆπ· after a rotation about a point π΄. What are the coordinates of π΄, and what is the degree measure of the rotation?
3. Find the line of reflection for the reflection that takes point π΄ to point π΄β². Find the image πβ² under this reflection.
4. Quinn tells you that the vertices of the image of quadrilateral πΆπ·πΈπΉ reflected over the line representing the equation π¦ = β 3/2π₯ + 2 are the following: πΆβ²(β2,3), π·β²(0,0), πΈβ²(β3,β3), and πΉβ²(β3,3). Do you agree or disagree with Quinn? Explain.
5. A translation takes π΄ to π΄β². Find the image πβ² and pre-image πβ²β² of point π under this translation. Find a vector that describes the translation.
6. The point πΆβ² is the image of point πΆ under a translation of the plane along a vector.
a. Find the coordinates of πΆ if the vector used for the translation is π΅π΄.
b. Find the coordinates of πΆ if the vector used for the translation is π΄π΅.

Exercises 7 - 9

1. A dilation with center π and scale factor π takes π΄ to π΄β² and π΅ to π΅β². Find the center π, and estimate the scale factor r.
2. Given a center π, scale factor π, and points π΄ and π΅, find the points π΄β² = π·π,π(π΄) and π΅β² = π·π,π(π΅). Compare length π΄π΅ with length π΄β²π΅β² by division; in other words, find π΄β²π΅β²/π΄π΅. How does this number compare to π?
3. Given a center π, scale factor π, and points π΄, π΅, and πΆ, find the points π΄β² = π·π,π(π΄), π΅β² = π·π,π(π΅), and πΆβ² = π·π,π(πΆ). Compare πβ π΄π΅πΆ with πβ π΄β²π΅β²πΆβ². 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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