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) 38° 75° 113° counterclockwise rotation with center (2,1)

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>

Compositions

Compositions

Compositions

Reflections over two parallel lines = One Translation

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.

Compositions

Reflections over two intersecting lines = One Rotation

Compositions

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 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)

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'

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)

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.

Remember: p. 654

Lesson Check

Lesson Check