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Geometry

Learn to identify rotations in the plane and apply rotation formulas to figures on the coordinate plane. Understand center of rotation, angle of rotation, and how to perform 90°, 180° rotations clockwise and counter-clockwise. Practice with examples and explore rotational symmetry concepts.

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Geometry

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  1. Geometry Rotations

  2. Goals • Identify rotations in the plane. • Apply rotation formulas to figures on the coordinate plane.

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

  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’

  5. A Rotation is an Isometry • Segment lengths are preserved. • Angle measures are preserved. • Parallel lines remain parallel. • Orientation is unchanged.

  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).

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

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

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

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

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

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

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

  14. Rotation Example B(-2, 4) Rotate ABC 90 clockwise. Formula (x, y)  (y, x) A(-3, 0) C(1, -1)

  15. Rotate ABC 90 clockwise. B(-2, 4) (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)

  16. Rotate ABC 90 clockwise. B(-2, 4) Check by rotating ABC 90. A’ B’ A(-3, 0) C’ C(1, -1)

  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.

  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.

  19. 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. k m P

  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

  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

  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.

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

  24. Does this figure have rotational symmetry? Yes, of 180.

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

  26. C B D A E H F G Rotating segments O

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

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

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

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

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

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

  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.

  34. Homework

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