1. Rotations California Standards for Geometry
16: Perform basic constructions
17: Prove theorems using coordinate geometry
22: Know the effect of rigid motions on figures in
the coordinate plane.

2. Properties of a 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.

3. Properties of a Rotation
If P is not C (the center of rotation),
then PC = P’C P xo P’ C

4. Properties of a Rotation
If P is C (the center of rotation),
then P = P’ P P’ C

5. Properties of a RotationQ’ Q P’ R S R’ P T’ T S’ C

6. identify and use rotationsQ’ Q P’ R S R’ P T’ T
88o S’ C

7. theorem 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’

8. theorem Three Cases are needed to prove that a rotation is an isometryQ P P’
Case 1: P, Q and C
are noncollinear C Q’

9. theorem Q
Case 2: P, Q, and C
are collinear Q’ P’ P C

10. theorem
Case 3: P and C are the
same point Q P C P’ Q’

11. Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’
Definition rotation
Definition rotation
+ prop of =

12. Case 1: P, Q and C are noncollinear Prove: PQ = P’Q’ C.P.C.T.C. Q P P’

13. and name the new coordinates
Example S R
Graph Quad PQRS
P(3, 1), Q(4, 0),
R(4, 3) S(2, 4)
and then rotate PQRS
180o counterclockwise
about (0, 0)
and name the new coordinates P
(-3, -1) P’ Q

14. and name the new coordinates
Example S R
Graph Quad PQRS
P(3, 1), Q(4, 0),
R(4, 3) S(2, 4)
and then rotate PQRS
180o counterclockwise
about (0, 0)
and name the new coordinates
(-4, 0) Q’ P
(-3, -1) P’ Q

15. and name the new coordinates
Example S R
Graph Quad PQRS
P(3, 1), Q(4, 0),
R(4, 3) S(2, 4)
and then rotate PQRS
180o counterclockwise
about (0, 0)
and name the new coordinates
(-4, 0) Q’ P
(-3, -1) P’ Q R’
(-4, -3)

16. and name the new coordinates
Example S R
Graph Quad PQRS
P(3, 1), Q(4, 0),
R(4, 3) S(2, 4)
and then rotate PQRS
180o counterclockwise
about (0, 0)
and name the new coordinates
(-4, 0) Q’ P
(-3, -1) P’ Q R’
(-4, -3) S’
(-2, -4)

17. and name the new coordinates
Example S R
Graph Quad PQRS
P(3, 1), Q(4, 0),
R(4, 3) S(2, 4)
and then rotate PQRS
180o counterclockwise
about (0, 0)
and name the new coordinates
(-4, 0) Q’ P
(-3, -1) P’ Q R’
(-4, -3) S’
(-2, -4)

18. theorem Reflection-Rotation Theorem
If two lines intersect, then a reflection in the first line
followed by a reflection in the second line is the same
as a rotation about the point of intersection. m A B B’ A’ P
B’’
A’’

19. theorem xo 2xo Reflection-Rotation Theorem
The angle of rotation is 2xo, where xo is the measure
of the acute or right angle formed by the two lines. m A
2xo xo B B’ A’ P
B’’
A’’

20. Example is reflected in line k to produce .
This triangle is the reflected in line m to produce
Describe the transformation k J’ J” K’ K” K L’ L”
90o clockwise rotation
45o J P L m

21. Definition 90o Rotational Symmetry
A figure that can be mapped onto itself by a
rotation of 180o or less.
90o

22. 120o Definition Rotational Symmetry
A figure that can be mapped onto itself by a
rotation of 180o or less.
120o

23. No rotational symmetry
Definition
Rotational Symmetry
A figure that can be mapped onto itself by a
rotation of 180o or less.
No rotational symmetry

24. Summary What are the properties of a rotation?
How are reflections and rotations related?
What does it mean when a figure has rotational symmetry?