Dilations as Transformations of the Plane


Related Topics:
Lesson Plans and Worksheets for Geometry
Lesson Plans and Worksheets for all Grades
More Lessons for Geometry
Common Core For Geometry




Share this page to Google Classroom

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 𝐴𝐡 to 𝐴′𝐡′. Find the image 𝑃′ of point 𝑃 under this rotation.
  2. In the diagram below, β–³ 𝐡′𝐢′𝐷′ is the image of β–³ 𝐡𝐢𝐷 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 𝐢𝐷𝐸𝐹 reflected over the line representing the equation 𝑦 = βˆ’ 3/2π‘₯ + 2 are the following: 𝐢′(βˆ’2,3), 𝐷′(0,0), 𝐸′(βˆ’3,βˆ’3), and 𝐹′(βˆ’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 𝐢′ is the image of point 𝐢 under a translation of the plane along a vector.
    a. Find the coordinates of 𝐢 if the vector used for the translation is 𝐡𝐴.
    b. Find the coordinates of 𝐢 if the vector used for the translation is 𝐴𝐡.



Exercises 7 - 9

  1. A dilation with center 𝑂 and scale factor π‘Ÿ takes 𝐴 to 𝐴′ and 𝐡 to 𝐡′. Find the center 𝑂, and estimate the scale factor r.
  2. Given a center 𝑂, scale factor π‘Ÿ, and points 𝐴 and 𝐡, find the points 𝐴′ = 𝐷𝑂,π‘Ÿ(𝐴) and 𝐡′ = 𝐷𝑂,π‘Ÿ(𝐡). Compare length 𝐴𝐡 with length 𝐴′𝐡′ by division; in other words, find 𝐴′𝐡′/𝐴𝐡. How does this number compare to π‘Ÿ?
  3. Given a center 𝑂, scale factor π‘Ÿ, and points 𝐴, 𝐡, and 𝐢, find the points 𝐴′ = 𝐷𝑂,π‘Ÿ(𝐴), 𝐡′ = 𝐷𝑂,π‘Ÿ(𝐡), and 𝐢′ = 𝐷𝑂,π‘Ÿ(𝐢). Compare π‘šβˆ π΄π΅πΆ with π‘šβˆ 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.

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

Fraction Concoction Game



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