9.4 Compositions of Transformations

Objective: Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines.

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.

Definition An isometry is a transformation that preserves distance. Translations, reflections and rotations are isometries.

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.

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.

Compositions We can perform more than one transformation to any single point, line, plane or figure. This is what we call compositions of transformations.

Two translations = One translation

Two rotations, same center equal One rotation

Find a single transformation for a 75° counterclockwise rotation with center (2,1) followed by a 38° counterclockwise rotation with center (2,1) 113° counterclockwise rotation with center (2,1) 75°38°

Find a single transformation equivalent to a translation with vector followed by a translation with vector. These Translations are equal to the Translation with vector

Compositions

Reflections over two parallel lines = One Translation

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.

Compositions

Reflections over two intersecting lines = One Rotation

Compositions

Glide Reflections

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 1translation 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 2reflection 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)

A.R' B.S' C.T' D.U' 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)?

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 1translation 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 2rotation 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)

A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown. Step 1A 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. The new brick is in place. Step 3

p. 654 Remember:

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