1. Bell Ringer
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2. Geometry
Rotations

3. Goals Identify rotations in the plane.
Apply rotation formulas to figures on the coordinate plane.
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4. Rotation
A transformation in which a figure is turned about a fixed point, called the center of rotation.
Center of Rotation
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5. 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’
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6. A Rotation is an Isometry
Segment lengths are preserved.
Angle measures are preserved.
Parallel lines remain parallel.
Orientation is unchanged.
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7. 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).
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8. 90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2)
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9. Rotate (-3, -2) 90 clockwise
Formula
(x, y)  (y, x)
A’(-2, 3)
(-3, -2)
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10. 90 counter-clockwise rotation
Formula
(x, y)  (y, x)
A’(2, 4)
A(4, -2)
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11. Rotate (-5, 3) 90 counter-clockwise
Formula
(x, y)  (y, x)
(-5, 3)
(-3, -5)
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12. 180 rotation
Formula
(x, y)  (x, y)
A’(4, 2)
A(-4, -2)
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13. Rotate (3, -4) 180 Formula (x, y)  (x, y) (-3, 4) (3, -4)
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14. Rotation Example Draw a coordinate grid and graph: A(-3, 0) B(-2, 4)
Draw ABC
A(-3, 0)
C(1, -1)
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15. Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x)
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16. 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)
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17. Rotate ABC 90 clockwise.
Check by rotating ABC 90. A’ B’
A(-3, 0) C’
C(1, -1)
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18. 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.
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19. 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.
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20. Does this figure have rotational symmetry?
The hexagon has rotational symmetry of 60.
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21. Does this figure have rotational symmetry?
Yes, of 180.
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22. Does this figure have rotational symmetry?
90
180
270
360
No, it required a full 360 to map onto itself.
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23. Rotating segments A B C D E F G H O
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24. Rotating AC 90 CW about the origin maps it to _______.CE A B C D E F G H O
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25. Rotating HG 90 CCW about the origin maps it to _______.FE A B C D E F G H O
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26. Rotating AH 180 about the origin maps it to _______.ED A B C D E F G H O
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27. Rotating GF 90 CCW about point G maps it to _______.GH A B C D E F G H O
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28. Rotating ACEG 180 about the origin maps it to _______.
EGAC C G A B C D E F G H A E O
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29. Rotating FED 270 CCW about point D maps it to _______.
BOD A B C D E F G H O
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30. 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.
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31. Homework
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