Pearson Unit 2 Topic 8: Transformational Geometry 8-8: Other Non-Rigid Transformations 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)(E) Create and use representations to organize, record, and communicate mathematical ideas. (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(F) Analyze mathematical relationships to connect 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. (3)(C) Identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane.

VOCABULARY Compression—a transformation that decreases the distance between corresponding points of a figure and a line Stretch—a transformation that increases the distance between corresponding points of a figure and a line

Answer these questions for each of the Discoveries #1-4: Is the transformation rigid or non-rigid? Describe the transformation as one of these: Vertical stretch Horizontal stretch Vertical compression Horizontal compression

Discovery #1: Graph ΔABC with A(0, 2), B(-4, -1), C(2, 1). Apply the motion rule (x, y) (x, 3y). Is the result a rigid transformation? A’(0, 6), B’(-4, -3), C’(2, 3). It is not rigid. The result is called a vertical stretch.

Discovery #2: Graph ΔABC with A(1, 2), B(-2, -5), C(3, -1). Apply the motion rule (x, y) (2x, y). Is the result a rigid transformation? A’(2, 2), B’(-4, -5), C’(6, -1). It is not rigid. The result is called a horizontal stretch.

Discovery #3: Graph ΔABC with A(6, 4), B(-6, -3), C(4, -5). Apply the motion rule (x, y) (½x, y). Is the result a rigid transformation? A’(3, 4), B’(-3, -3), C’(2, -5). It is not rigid. The result is called a horizontal compression.

Discovery #4: Graph ΔABC with A(6, 4), B(-5, -4), C(2, -6). Apply the motion rule (x, y) (x, ¾y). Is the result a rigid transformation? A’(6, 3), B’(-5, -3), C’(2, -4.5). It is not rigid. The result is called a vertical compression.

Example #1: Is ΔSTW ΔS’T’W’ a vertical or horizontal compression, or a vertical or horizontal stretch? Write a motion rule that maps ΔSTW to ΔS’T’W’. This is a vertical compression. Compare S’T’ to ST to find the scale factor. 2 8 = 1 4 (x, y) (x, ¼y)

Example #2: ΔABC has vertices A(-2, -2), B(1, 2), C(0, -3). Determine the vertices of the image of ΔABC after a dilation with scale factor 1.5 centered at the origin, followed by the transformation (x, y) (2x, y). A’(-3, -3), B’(1.5, 3), C’(0, -4.5) A”(-6, -3), B”(3, 3), C”(0, -4.5)

Example #3: Stretch ABCD using the rule (x, y) (x, 1.5y). An architect’s plan for a city park is shown at the left below. The mayor of the city asks that the swimming pool be longer, but not wider, and wants to move it to the other end of the park. Describe a sequence of transformations that will move the pool to the outlined location on the architect’s plan. Stretch ABCD using the rule (x, y) (x, 1.5y). Then translate using the rule (x, y) (x + 10, y + 5).