Rotations California Standards for Geometry 16: Perform basic constructions 17: Prove theorems using...
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Rotations California Standards for Geometry 16: Perform basic constructions 17: Prove theorems using coordinate geometry 22: Know the effect of rigid
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Transcript of Rotations California Standards for Geometry 16: Perform basic constructions 17: Prove theorems using...
Rotations
California Standards for Geometry16: Perform basic constructions17: Prove theorems using coordinate geometry22: Know the effect of rigid motions on figures in the coordinate plane.
Properties of a Rotation• Rotation
– 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 point and its image form the ANGLE OF ROTATION.– Rotations can be clockwise or counterclockwise.
'm PCP x
C
If P is not C (the center of rotation),
then PC = P’C
P P’xo
Properties of a Rotation
Properties of a Rotation
If P is C (the center of rotation), then P = P’
P
C
P’
identify and use rotations
C
P
T
Q
RS
T’
P’
Q’
R’
S’
88o
Rotation Theorem A rotation is an isometry
to prove this theorem, you must show that a rotation keeps segment lengths from the
preimage to the image
this means that AB = A’B’
theorem
theorem• Reflection-Rotation Theorem
If two lines intersect, then a reflection in the first linefollowed by a reflection in the second line is the sameas a rotation about the point of intersection.
B
Am
P
B’ A’
B’’ A’’
• Reflection-Rotation TheoremThe angle of rotation is 2xo, where xo is the measure of the acute or right angle formed by the two lines.
theorem
xoB
A
P
B’ A’
B’’ A’’
2xo
m
Example
is reflected in line k to produce . This triangle is the reflected in line m to produce Describe the transformation
JKL ' ' 'J K L" " "J K L
" " "JKL J K L k
m
J
K
L
J’
K’
L’
J”
K”
L”
45o
P
90o clockwise rotation
• Rotational SymmetryA figure that can be mapped onto itself by a rotation of 180o or less.
Definition
90o
• Rotational SymmetryA figure that can be mapped onto itself by a rotation of 180o or less.
Definition
120o
• Rotational SymmetryA figure that can be mapped onto itself by a rotation of 180o or less.
Definition
No rotational symmetry
Summary
• What are the properties of a rotation?• How are reflections and rotations related?• What does it mean when a figure has rotational
symmetry?
