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 3
Rotations in the Coordinate
Plane
Hexagon 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)
EXAMPLE 3
Triangle PQR is shown below. What is the image of point Q
after a 90° counterclockwise rotation about the origin?
A. (–5, –4)
B. (–5, 4)
C. (5, 4)
D. (4, –5)
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
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'
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
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''?
A. (–3, –1)
B. (–6, –1)
C. (1, 6)
D. (–1, –6)
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
Step 1
Reflect a Figure in Two Lines
Reflect EFGH in line p.
EXAMPLE 3
Step 2
Reflect a Figure in Two Lines
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
Copy and reflect figure ABC in line
s and then line t. Then describe a
single transformation that maps
ABC onto A''B''C''.
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''.
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. What transformation must occur to the
brick at point M to further complete the
pattern shown here?
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.
EXAMPLE 4
B. What transformation must occur to the
brick at point M to further complete the
pattern shown here?
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.