Pearson Unit 2 Topic 8: Transformational Geometry 8-5: Compositions of Rigid Transformations You will need a piece of patty paper for today’s notes. Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (3)(A) Describe and perform transformations of figures in a plane using coordinate notation. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. (3)(B) Determine the image or pre-image of a given two-dimensional figures under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane.

Discovery #1 Given: 2 parallel lines and a triangle. Reflect the triangle across the first parallel line, followed by another reflection across the second parallel line. d A composition of reflections across 2 parallel lines is a translation. If d is the distance between the 2 parallel lines, the distance from the pre-image to the final image is 2d.

Discovery #2 Given: 2 intersecting lines and a triangle. Reflect the triangle across the line m, followed by another reflection across line n. m x n A composition of reflections across 2 intersecting lines is a rotation. If x is the angle between the 2 intersecting lines, the angle of rotation from the pre-image to the final image is 2x.

If you reflect first and then translate, is the final image in the same place? Yes.

Example #1 Find the image of each letter after the transformation 𝑅 𝑛 ° 𝑅 𝑚 . Note: remember that you do 𝑅 𝑚 first, followed by 𝑅 𝑛 .

Example #2 Find the image of each letter after the transformation 𝑅 𝑛 ° 𝑅 𝑚 . Note: remember that you do 𝑅 𝑚 first, followed by 𝑅 𝑛 .

Example #3 Graph ΔBPN with B(-4, -3), P(0, 2), and N(3, -1) and its image after the given transformation of (𝑅 𝑦=1° 𝑇 2, 0 )(ΔBPN). Remember to do the translation first, then do the reflection.

Example #4 Graph ΔBPN with B(-4, -3), P(0, 2), and N(3, -1) and its image after the given transformation of (𝑅 𝑦=0° 𝑟 180, 𝑂 )(ΔBPN). Remember to do the rotation first, and then the reflection.