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Ch 9-3
• rotation
• center of rotation
• angle of rotation
• rotational symmetry
• invariant points
• direct isometry
• indirect isometry
• Draw rotated images using the angle of rotation.
• Identify figures with rotational symmetry.
Rotations
A transformation in which a figure is turned about a fixed point.The fixed point is the Center of RotationRays drawn from the center of rotation to a point and its image form an angle called the Angle of Rotation.
Hi
Watch when this rectangle is rotated by a given angle measure.
Center of
Rotation
Hi
Angle of Rotation
Center of
Rotation
A. A
B. B
C. C
D. D0% 0%0%0%
A. For the following diagram, which description best identifies the rotation of triangle ABC around point Q?
A. 20° clockwise
B. 20° counterclockwise
C. 90° clockwise
D. 90° counterclockwise
Rotations• A composite of two reflections over two
intersecting lines
• The angle of rotation is twice the measure of the angle b/t the two lines of reflection
• Coordinate Plane rotation
Rotating about the origin
Reflections in Intersecting Lines
Find the image of parallelogram WXYZ under reflections in line p and then line q.
Answer: Parallelogram W''X''Y''Z'' is the image of parallelogram WXYZ under reflections in line p and q.
First reflect parallelogram WXYZ in line p. Then label the image W'X'Y'Z'.
Next, reflect the image in line q. Then label the image W''X''Y''Z''.
1. A
2. B
3. C
0% 0%0%
A. blue Δ
B. green Δ
C. neither
In the following diagram, which triangle is the image of ΔABC under reflections in line m and then line n.
Coordinate Plane RotationRotating about the origin• Clockwise vs.Counterclockwise• 90o Quarter turn• 180o Half turn (clockwise or counterclockwise)
• 270o Three quarter turn
Big Hint!!!If you need to rotate a shape about the origin,• TURN THE PAPER• Write down the new coordinates• Turn the paper back and graph the rotated points.
Example #1• Rotate ABC 90o clockwise about the origin.
Turn the paper
(90o clockwise)
Write the new coordinates
A’ (2, 4)
B’ (4, 1)
C’ (-1, 3)
Turn the paper back and graph the rotated points
4
2
-2
-5 5
A
B
C
A’
B’C’
4
2
-2
-4
-5 5
A
B
C
Example #2• Rotate ABC 180o about the origin.
Turn the paper (180o)
Write the new coordinates
A’ (4, -2)
B’ (1, -4)
C’ (3, 1)
Turn the paper back and graph the rotated points
A’
B’
C’
Rotational Symmetry• A figure has rotational symmetry if it can be
mapped onto itself by a rotation of 180º or less.
– Equilateral Triangle
– Square
– Most regular polygonsA B
CD
An equilateral triangle maps onto itself every 120 degrees of rotation.
There are 3 rotations (<360 degrees) where the triangle maps onto itself.
1203360 magnitude of symmetry
The equilateral triangle has rotational symmetry of order = 3.
An regular pentagon has an order of 5.
725360 magnitude of symmetry
1
2
34
5 123
4
5
12
3
45
1
23
4
5
1
2
345
1
2
34
5
Draw a Rotation
• Use a protractor to measure a 45° angle counterclockwise with as one side. Extend the other side to be longer than AR.
• Draw a segment from point R to point A.
• Locate point R' so that AR = AR'.
A. Rotate quadrilateral RSTV 45° counterclockwise about point A.
• Repeat this process for points S, T, and V.
• Connect the four points to form R'S'T'V'.
Draw a Rotation
Quadrilateral R'S'T'V' is the image of quadrilateral RSTV under a 45° counterclockwise rotation about point A.
Answer:
Draw a RotationB. Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Draw the image of DEF under a rotation of 115° clockwise about the point G(–4, –2).
First draw ΔDEF and plot point G.
Use a protractor to measure a 115° angle clockwise with as one side.
Use a compass to copy onto Name the segment
Draw
Repeat with points E and F.
Draw a segment from point G to point D.
Draw a Rotation
ΔD'E'F' is the image of ΔDEF under a 115° clockwise rotation about point G.
Answer:
A. A
B. B
C. C
D. D
0% 0%0%0%
B. Triangle ABC has vertices A(1, –2), B(4, –6), and C(1, –6). Draw the image of ΔABC under a rotation of 70° counterclockwise about the point M(–1, –1).
A. B.
C. D.
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