Pearson Unit 2 Topic 8: Transformational Geometry 8-3: Rotations 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)(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. (6)(C) Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

Example #1 Identify the transformation. Then use arrow notation to describe the transformation. rotation; ∆XYZ  ∆X’Y’Z’

Example #2 Describe the transformation. 90° rotation counterclockwise about the origin ∆ABC  ∆A’B’C’

Example #3 Rotate ∆ABC by 90° about the origin. Remember that a rotation is always counterclockwise unless stated otherwise. B’ Graph the preimage and image. C C’ B A’ The rotation of (x, y) is (–y, x). A(2, –1) A’ (1, 2) A B(4, 1) B’ (–1, 4) C(3, 3) C’ (–3, 3)

Example #4 Rotate ΔJKL with vertices J(3, 5), K(4, –5), and L(–1, 6) by 180° about the origin. L Graph the preimage and image. J’ K’ L’ J The rotation of (x, y) is (–x, –y). J(3, 5) J’ (–3, -5) K(4, –5) K’ (–4, 5) K L(–1, 6) L’ (1, –6)

Example #5 The Millenium Wheel, also known as the London Eye, contains 32 observation cars. Determine the angle of rotation that will bring Car 3 to the position of Car 18. First: 360/32 = 11.25 Second: Car 18 – Car 3 = 15 Third: (15)(11.25) = 168.75

Example #6 For center of rotation P, does an x rotation followed by a y rotation give the same image as a y rotation followed by an x rotation? The final result is in the same location, so the order did not matter.

Example #7 First: Circumscribe a circle. Second: Draw the radii. Point O is the center of a regular nonagon. Describe a rotation that maps H to C. First: Circumscribe a circle. Second: Draw the radii. Third: Find the measure of the central angles using 360/9 = 40. Fourth: from H to C is 5 rotations of 40. Fifth: 5(40) = 200.

Example #8 Rotate ΔABC 60 from point P. 1. Draw a segment from A to P. Measure length. 2. Draw a 60 angle with vertex P and side AP. 3. Use a compass to construct PA  PA’. 4. Locate B’ and D’ in a similar manner. 5. Draw ΔA’B’C’. 6. Or after steps #1-2, you can use patty paper to rotate the triangle.

Example #9 Graph ABCD. Rotate ABCD 90 about point A. Rotate ABCD 180 about point B. Rotate ABCD 90 about point C. Rotate ABCD 90 about the midpoint of CD. A D B C