Transcript Rotations

Ch 9-3
• Draw rotated images
using the angle of
rotation.
• Identify figures with
rotational symmetry.
• rotation
• center of rotation
• angle of rotation
• rotational symmetry
• invariant points
• direct isometry
• indirect isometry
Rotations
A transformation in which a figure is turned
about a fixed point.
The fixed point is the Center of Rotation
Rays drawn from the center of rotation to a
point and its image form an angle called the
Angle of Rotation.
Watch when this rectangle is rotated by a given
angle measure.
Hi
Center
of
Angle of
Rotation
Hi
Center
of
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
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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.
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''.
Answer: Parallelogram W''X''Y''Z'' is the image of
parallelogram WXYZ under reflections in line p
and q.
In the following diagram, which triangle
is the image of ΔABC under reflections
in line m and then line n.
A. blue Δ
B. green Δ
C. neither
A
B
C
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Coordinate Plane Rotation
Rotating 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
B4
(90o clockwise)
Write the new
coordinates
A’ (2, 4)
B’ (4, 1)
A
C’
A’
2
B’
-5
5
C
-2
C’ (-1, 3)
Turn the paper back and graph the rotated points
Example #2
• Rotate ABC 180o about the origin.
Turn the paper
B
(180o)
4
Write the new
coordinates
A
2
A’ (4, -2)
C’
-5
B’ (1, -4)
C’ (3, 1)
Turn the paper back and
graph the rotated points
5
C
-2
B’
-4
A’
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 polygons
A
B
D
C
There are 3 rotations (<360 degrees) where
the triangle maps onto itself.
The equilateral triangle
has rotational symmetry
of order = 3.
360   3  120 
magnitude of
symmetry
An equilateral triangle maps onto itself
every 120 degrees of rotation.
An regular pentagon has an order of 5.
1
5
2
4
3
1
5
2
4
360   5  72
magnitude of
symmetry
3
Draw a Rotation
A. Rotate quadrilateral RSTV 45° counterclockwise
about point A.
• Draw a segment from point R to
point A.
• Use a protractor to measure a
45° angle counterclockwise with
as one side. Extend the other
side to be longer than AR.
• Locate point R' so that AR = AR'.
• 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 Rotation
B. 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.
Draw a segment from point G to
point D.
Use a protractor to measure a 115°
angle clockwise with
as one side.
Draw
Use a compass to copy
onto
Name the segment
Repeat with points E and F.
Draw a Rotation
ΔD'E'F' is the image of ΔDEF under a 115° clockwise
rotation about point G.
Answer:
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).
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