1 / 34

# Geometry

Geometry. Rotations. Goals. Identify rotations in the plane. Apply rotation formulas to figures on the coordinate plane. Rotation. A transformation in which a figure is turned about a fixed point, called the center of rotation. Center of Rotation. Rotation.

Télécharger la présentation

## Geometry

An Image/Link below is provided (as is) to download presentation Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

### Presentation Transcript

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

More Related