1. Geometry
Rotations

2. Goals Identify rotations in the plane.
Apply rotation formulas to figures on the coordinate plane.
4/23/2017

3. Rotation
A transformation in which a figure is turned about a fixed point, called the center of rotation.
Center of Rotation
4/23/2017

4. Rotation
Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. G
90
Center of Rotation G’
4/23/2017

5. A Rotation is an Isometry
Segment lengths are preserved.
Angle measures are preserved.
Parallel lines remain parallel.
Orientation is unchanged.
4/23/2017

6. Rotations on the Coordinate Plane
Know the formulas for:
90 rotations
180 rotations
clockwise & counter-clockwise
Unless told otherwise, the center of rotation is the origin (0, 0).
4/23/2017

7. 90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2)
4/23/2017

8. Rotate (-3, -2) 90 clockwise
Formula
(x, y)  (y, x)
A’(-2, 3)
(-3, -2)
4/23/2017

9. 90 counter-clockwise rotation
Formula
(x, y)  (y, x)
A’(2, 4)
A(4, -2)
4/23/2017

10. Rotate (-5, 3) 90 counter-clockwise
Formula
(x, y)  (y, x)
(-5, 3)
(-3, -5)
4/23/2017

11. 180 rotation
Formula
(x, y)  (x, y)
A’(4, 2)
A(-4, -2)
4/23/2017

12. Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4)
4/23/2017

13. Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4)
Draw ABC
A(-3, 0)
C(1, -1)
4/23/2017

14. Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x)
4/23/2017

15. Rotate ABC 90 clockwise.
(x, y)  (y, x)
A(-3, 0)  A’(0, 3)
B(-2, 4)  B’(4, 2)
C(1, -1)  C’(-1, -1) A’ B’
A(-3, 0) C’
C(1, -1)
4/23/2017

16. Rotate ABC 90 clockwise.
Check by rotating ABC 90. A’ B’
A(-3, 0) C’
C(1, -1)
4/23/2017

17. Rotation Formulas 90 CW (x, y)  (y, x) 90 CCW (x, y)  (y, x)
180 (x, y)  (x, y)
Rotating through an angle other than 90 or 180 requires much more complicated math.
4/23/2017

18. Compound Reflections
If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P.
4/23/2017

19. Compound Reflections k m P
If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. k m P
4/23/2017

20. Compound Reflections
Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m. k m
45
90 P
4/23/2017

21. Compound Reflections
The amount of the rotation is twice the measure of the angle between lines k and m. k m x
2x P
4/23/2017

22. Rotational Symmetry
A figure can be mapped onto itself by a rotation of 180 or less.
45
90
The square has rotational symmetry of 90.
4/23/2017

23. Does this figure have rotational symmetry?
The hexagon has rotational symmetry of 60.
4/23/2017

24. Does this figure have rotational symmetry?
Yes, of 180.
4/23/2017

25. Does this figure have rotational symmetry?
90
180
270
360
No, it required a full 360 to map onto itself.
4/23/2017

26. Rotating segments A B C D E F G H O
4/23/2017

27. Rotating AC 90 CW about the origin maps it to _______.CE A B C D E F G H O
4/23/2017

28. Rotating HG 90 CCW about the origin maps it to _______.FE A B C D E F G H O
4/23/2017

29. Rotating AH 180 about the origin maps it to _______.ED A B C D E F G H O
4/23/2017

30. Rotating GF 90 CCW about point G maps it to _______.GH A B C D E F G H O
4/23/2017

31. Rotating ACEG 180 about the origin maps it to _______.
EGAC C G A B C D E F G H A E O
4/23/2017

32. Rotating FED 270 CCW about point D maps it to _______.
BOD A B C D E F G H O
4/23/2017

33. Summary
A rotation is a transformation where the preimage is rotated about the center of rotation.
Rotations are Isometries.
A figure has rotational symmetry if it maps onto itself at an angle of rotation of 180 or less.
4/23/2017

34. Homework
4/23/2017