September 30, 2016
September 30, 2016
September 30, 2016
Aim #13: How do we rotate figures?
Do Now:
A counterclockwise rotation
turns to theleft and has a
positive degree measure.
CC Geometry H
A clockwise rotation
turns to the right and has
a negative degree measure.
Using the stick figure pre-image:
a) Sketch the image after
0
a 90 counterclockwise
rotation.
b) Sketch the image
c) Sketch the image
0
0
0
after a -90 (90
after a 180 rotation.
clockwise) rotation.
A rotation is a transformation in which a figure is turned about a fixed point
called the center of rotation. Rays drawn from the center of rotation to a preimage point to its image form the angle of rotation.
0
Rotate ΔABC 90 counterclockwise
about the origin.
0
Rotate ΔABC 90 clockwise
about the origin.
C'
A'
A( ­6 , 1 )
A'( ­1 , ­6 )
C
A
B
B( ­2 , 1 )
B'( ­1 , ­2 )
B'
C'
C
A
B
B'
C( ­6 , 3 )
C'( ­3 , ­6 )
A'
0
Rotate ΔABC 180
counterclockwise about the origin.
0
Rotate ΔABC 180 clockwise
about the origin. same as CCW!
A( ­6 , 1 )
A'( 6 , ­1 )
C
A
B
B'
B( ­2 , 1 )
A' B'( 2 , ­1 )
C'
C( ­6 , 3 )
C'( 6 , ­3 )
C
A
B
September 30, 2016
The symbolic notation for a rotation is RC,θ(P)
where R stands for Rotation, C is the center point of rotation,
θ is the angle measure of the rotation, and P is the point to be rotated.
0
0
For 0 < θ < 180 , the rotation of θ degrees around the center C is the
transformation RC,θ defined as follows:
1) For the center point C, RC ,θ(C) = C
θ
C
Q
2) For any other point P, RC,θ(P) = Q , where CQ = CP and m≮PCQ = θ.
θ
C
o
P
3) A rotation of 0 around the center C is the identity transformation
defined by RC,0 (P) = P.
θ = 00
P = preimage/image
C
Exercises:
0
1. Which of the following is equivalent to rotating 67 CCW?
0
0
0
0
A. 23 CW
B. 113 CW
C. 113 CCW
D. 293 CW
#2-4 You will need to use a protractor to measure angles.
2. To determine the angle of rotation, you measure
the angle formed by connecting corresponding
vertices to the center point of rotation.
Measure ≮AD'A'. ______ What happened to D ?
_____________________________________
Draw ≮BD'B'. Do you get the same angle measure?____
What about ≮CD'C'?_____
3. Find the angle of rotation. Name it three ways.
September 30, 2016
0
4. Using your compass and straightedge only, draw a 60 CCW rotation of ΔDEF
about point P.
E
P
D
F
5. Let‛s move on to finding the center of rotation. Examine the figures below.
Notice the points A, B, A', and B'.Follow the directions below to locate the center
of rotation. The pre-image is solid.
a) Draw a segment connecting points A and A'.
b) Using a compass and straightedge, construct the perpendicular bisector of AA'.
c) Draw a segment connecting points B and B'.
d) Construct the perpendicular bisector of BB'.
e) The point of intersection of the two perpendicular bisectors is the center of
rotation. Label this point P.
f) Measure angles ≮APA' and ≮BPB'. Did you obtain the same degree measure for
the angle of rotation? Name it three ways.
September 30, 2016
6. Find the center of rotation, P, and the angle of rotation for the transformation
below that maps ABCD onto A'B'C'D'. Name the angle of rotation three ways.
B
D
C
C'
A
D'
B'
A'
7. Find the center of rotation, P, and angle of rotation for the pair of figures.
Name the angle of rotation three ways.
Angle BPB': 120o CW, ­120o, 2400
P
September 30, 2016
0
8. Rotate the quadrilateral 60 CCW around point P using a compass and
straightedge only.
C
B'
B
D
C'
A'
A
D'
P
9. Using the triangle below,
a) locate the midpoint of the longest side using your compass.
b) rotate the triangle 180° around the midpoint of the longest side.
What figure have you formed?
A
C
B
Let's Sum it Up!!
• Rotations preserve angle measure and lengths of segments.
• They are rigid motions (isometries).
• A rotation carries segments onto segments of equal length.
• A rotation carries angles onto angles of equal measure.
September 30, 2016
P
b)
P
a)
c)
CC Geometry H
HW #13
P
Name____________________
Date____________________
0
0
0
1. Describe each image as a rotation of 90 , 180 or -90 of P.
2. Find the center of rotation, P, and angle of rotation for the pair of figures.
Name the angle of rotation three ways.
0
3. Rotate quadrilateral ABCD 60 CCW around point E using a straightedge and
compass, only.
OVER
September 30, 2016
4a) Using a compass and straightedge, construct and label ≮RST with measure 45°.
b) A gain using only a compass and straightedge, construct and label ≮R'S'T', the
image of ≮RST following a 180°rotation around S.
c) What type of angles are the pre-image and image?
S
T
5a) Construct equilateral ΔABC.
b) Locate the midpoint, M, of one side using your compass and straightedge.
c) Construct ΔA'B'C', the image of ΔABC after a rotation of 180° around M.
A
B
Review
6. Using the figure to the right, what rigid motion(s)
a) map ΔBCA onto ΔEFD?
B
H
E
b) ΔBCA onto ΔECD
F
C
c) BC onto FD
d) map ΔACD onto ΔGFH?
A
D
G