CHAPTER 9.3 AND 9.4

Rotations and Compositions of Transformations

ROTATIONS

• A rotation is a turn that moves every point of an image through a specified angle and direction about a fixed point.

CONCEPT

ROTATE 90°

• Graph the point A

(-2, 4)

• Graph the image of A’ under a rotation of 90° counterclockwise about the origin

ROTATE 180°

• Graph the point B

(2, -5)

• Graph the image of B’ under a rotation of 180° counterclockwise about the origin

ROTATE 270°

• Graph the point C

(-4, -6)

• Graph the image of C’ under a rotation of 270° counterclockwise about the origin

EXAMPLE 3Hexagon DGJTSR is shown below. What is the image of point T after a 90 counterclockwise rotation about the origin?

A (5, –3)B (–5, –3)C (–3, 5)D (3, –5)

Rotations in the Coordinate Plane

EXAMPLE 3

A. (–5, –4)

B. (–5, 4)

C. (5, 4)

D. (4, –5)

Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin?

Rotations in the Coordinate Plane

Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 90° counterclockwise about the origin.

Rotations in the Coordinate Plane

Line segment XY has vertices X(0, 4) and Y(5, 1). Graph XY and its image after a rotation of 270° counterclockwise about the origin.

COMPOSITION OF TRANSFORMATIONS

• When a transformation is applied to a figure and then another transformation is applied to its image, the result is called a composition of transformations.

CONCEPT

EXAMPLE 1 Graph a Glide Reflection

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.

EXAMPLE 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)?

EXAMPLE 2 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.

EXAMPLE 2

A. (–3, –1)

B. (–6, –1)

C. (1, 6)

D. (–1, –6)

ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔTUV and its image after a translation along 3, 1 and a rotation 180° about the origin. What are the new coordinates of L''?

CONCEPT

CONCEPT

EXAMPLE 3 Reflect a Figure in Two Lines

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''.

EXAMPLE 3 Reflect a Figure in Two Lines

Step 1 Reflect EFGH in line p.

EXAMPLE 3 Reflect a Figure in Two Lines

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.

EXAMPLE 3

A. ABC is reflected across lines and translated down 2 inches.

B. ABC is translated down 2 inches onto A''B''C''.

C. ABC is translated down 2 inches and reflected across line t.

D. ABC is translated down 4 inches onto A''B''C''.

Copy and reflect figure ABC in line s and then line t. Then describe a single transformation that maps ABC onto A''B''C''.

EXAMPLE 4 Describe Transformations

A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

EXAMPLE 4 Describe Transformations

B. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

EXAMPLE 4

A. The brick must be rotated 180° counterclockwise about point M.

B. The brick must be translated one brick width right of point M.

C. The brick must be rotated 90° counterclockwise about point M.

D. The brick must be rotated 360° counterclockwise about point M.

A. What transformation must occur to the brick at point M to further complete the pattern shown here?

EXAMPLE 4

A. The two bricks must be translated one brick length to the right of point M.

B. The two bricks must be translated one brick length down from point M.

C. The two bricks must be rotated 180° counterclockwise about point M.

D. The two bricks must be rotated 90° counterclockwise about point M.

B. What transformation must occur to the brick at point M to further complete the pattern shown here?