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9.4 Composition of Transformations
Then: You drew reflections, translations, and rotations. Now: 1. Draw glide reflections and other compositions of isometries in the coordinate plane. 2. Draw compositions of reflections in parallel and intersecting lines.
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9.4 Composition of Transformations
I. Definition: The resulting transformation when a transformation is applied to a figure and then another transformation is applied to its image. Artwork by Escher
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9.4 Composition of Transformations
II. Types of Composition of Transformations A. Glide reflection: the composition of a translation followed by a reflection in a line parallel to the translation vector.
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9.4 Composition of Transformations
B. Graph a Glide Reflection 1. Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis. B’ (_____, _____) B” (_____, _____) G’ (_____, _____) G” (_____, _____) T’ (_____, _____) T” (_____, _____) S’ (_____, _____) S” (_____, _____)
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9.4 Composition of Transformations
B. Graph a Glide Reflection 2. Triangle XYZ has vertices X(6, 5), Y(7, –4) and Z(5, –5). Graph △XYZ and its image after a translation along 〈1, 2〉 and a reflection in the y-axis. X’ (_____, _____) X” (_____, _____) Y’ (_____, _____) Y” (_____, _____) Z’ (_____, _____) Z” (_____, _____)
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9.4 Composition of Transformations
3. From Problem #2, XYZ X’Y’Z’ and X’Y’Z X”Y”Z”, so by Transitive Property of Congruence, then XYZ X”Y”Z”. 4. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane. 5. Theorem 9.1 Composition of Isometries The composition of two (or more) isometries is an isometry.
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9.4 Composition of Transformations
C. Graph Other Compositions of Isometries 1. ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph TUV and its image after a translation along –1 , 5 and a rotation 180° about the origin. T’ (_____, _____) T” (_____, _____) U’ (_____, _____) U” (_____, _____) V’ (_____, _____) V” (_____, _____)
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9.4 Composition of Transformations
C. Graph Other Compositions of Isometries 2. ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔJKL and its image after a translation along 3, 1 and a rotation 90° about the origin. J’ (_____, _____) J” (_____, _____) K’ (_____, _____) K” (_____, _____) L’ (_____, _____) L” (_____, _____)
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9.4 Composition of Transformations
D. Composition of Two Reflections 1. Composition of two reflection in parallel lines is the same as a translation.
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9.4 Composition of Transformations
D. Composition of Two Reflections 2. Composition of two reflections in intersecting lines is the same as a rotation.
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9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern. 1. Carpet pattern:
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9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern. 2. Tile pattern:
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9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern. 3. Wallpaper pattern:
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9.4 Composition of Transformations
Assignment: P #8-16 evens, all, all, all, 38, all Graph: 8-16 evens, 25 & 27 (2 pieces of graph paper)
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