1. Transformations - Rotations
Geometry
Transformations - Rotations

2. Goals Identify rotations in the plane.
Apply rotation formulas to figures on the coordinate plane.
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3. 4-3 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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4. 4-3 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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5. 4-3 Rotation
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6. A Rotation is an Isometry
Segment lengths do not change.
Angle measures do not change.
Parallel lines remain parallel.
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7. 4-3 Rotation
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8. 4-3 Rotation
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9. Rotations on the Coordinate Plane
Know the formulas for:
90 rotations
180 rotations
270 rotations
clockwise & counter-clockwise
Unless told otherwise, the center of rotation is the origin (0, 0).
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10. 90 clockwise rotation Formula (x, y)  (y, x) A(-2, 4) A’(4, 2)
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11. Rotate (-3, -2) 90 CW Formula (x, y)  (y, x) A’(-2, 3) (-3, -2)
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12. 90 counter-clockwise rotation
Formula
(x, y)  (y, x)
A’(2, 4)
A(4, -2)
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13. Rotate (-5, 3) 90 counter-clockwise
Formula
(x, y)  (y, x)
(-5, 3)
(-3, -5)
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14. 180 clockwise rotation Formula (x, y)  (x, y) A’(4, 2) A(-4, -2)
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15. Rotate (3, -4) 180CW Formula (x, y)  (x, y) (-3, 4) (3, -4)
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16. 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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17. Rotation Example Rotate ABC 90 clockwise. Formula (x, y)  (y, x)
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18. 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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19. Rotate ABC 90 clockwise.
Check by rotating ABC 90. A’ B’
A(-3, 0) C’
C(1, -1)
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20. Rotation Formulas 90 CW (x, y)  (y, x) 90 CCW (x, y)  (y, x)
180 CW (x, y)  (x, y)
270 CCW (x, y)  (y, x)
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21. 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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22. Does this figure have rotational symmetry?
The hexagon has rotational symmetry of 60.
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23. Does this figure have rotational symmetry?
Yes, of 180.
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24. Does this figure have rotational symmetry?
90
180
270
360
No, it required a full 360 to map onto itself.
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25. 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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