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9.4 Compositions of Transformations
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Objective: Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines.
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Vocabulary: Composite photographs are made by superimposing one or more photographs. Morphing is a popular special effect in movies. It changes one image into another.
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Definition An isometry is a transformation that preserves distance.
Translations, reflections and rotations are isometries.
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Definition When a transformation is applied to a figure, and then another transformation is applied to its image, the result is called a composition of the transformations.
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The composition of two or more isometries – reflections, translations, or rotations results in an image that is congruent to its preimage. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane.
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Compositions We can perform more than one transformation to any single point, line, plane or figure. This is what we call compositions of transformations.
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Two translations = One translation
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Two rotations, same center equal
One rotation
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Find a single transformation for a 75° counterclockwise rotation with center (2,1) followed by a 38° counterclockwise rotation with center (2,1) 38° 75° 113° counterclockwise rotation with center (2,1)
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Find a single transformation equivalent to a translation with vector <−2, 7> followed by a translation with vector <9, 3>. These Translations are equal to the Translation with vector <-2+9, 7+3> <7, 10>
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Compositions
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Compositions
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Compositions
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Reflections over two parallel lines = One Translation
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Copy and reflect figure EFGH in line p and then line q
Copy and reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''. Step 1: Reflect EFGH in line p. Step 2: Reflect E'F'G'H' in line q. Answer: EFGH is transformed onto E''F''G''H'' by a translation down a distance that is twice the distance between lines p and q.
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Compositions
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Reflections over two intersecting lines = One Rotation
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Compositions
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Compositions
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Glide Reflections
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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. Step 1 translation along 5, 0 (x, y) → (x + 5, y) B(–3, 4) → B'(2, 4) G(–1, 3) → G'(4, 3) S(–4, 2) → S'(1, 2) T(–1, 1) → T'(4, 1) Step 2 reflection in the x-axis (x, y) → (x, –y) B'(2, 4) → B''(2, –4) G'(4, 3) → G''(4, –3) S'(1, 2) → S''(1, –2) T'(4, 1) → T''(4, –1)
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Quadrilateral RSTU has vertices R(1, –1), S(4, –2), T(3, –4), and U(1, –3). Graph RSTU and its image after a translation along –4, 1 and a reflection in the x-axis. Which point is located at (–3, 0)? A. R' B. S' C. T' D. U'
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Graph Other Compositions of Isometries
ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin. Step 1 translation along –1 , 5 (x, y) → (x + (–1), y + 5) T(2, –1) → T'(1, 4) U(5, –2) → U'(4, 3) V(3, –4) → V'(2, 1) Step 2 rotation 180 about the origin (x, y) → (–x, –y) T'(1, 4) → T''(–1, –4) U'(4, 3) → U''(–4, –3) V'(2, 1) → V''(–2, –1)
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A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.
Step 1 A brick is copied and translated to the right one brick length. Step 2 The brick is then rotated 90° counterclockwise about point M, given here. Step 3 The new brick is in place.
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Remember: p. 654
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Lesson Check
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Lesson Check
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