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13.6 and 13.7

13.6 and 13.7. Rotations, Symmetry and Dilations. Goal Statement. w ill rotate figures and identify rotational symmetry w ill dilate figures in a coordinate plane. Can you identify the following terms?. r otation: center of rotation: angle of rotation:.

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13.6 and 13.7

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  1. 13.6 and 13.7 Rotations, Symmetry and Dilations

  2. Goal Statement • will rotate figures and identify rotational symmetry • will dilate figures in a coordinate plane

  3. Can you identify the following terms? • rotation: • center of rotation: • angle of rotation:

  4. Can you identify the following terms? • rotation: a transformation where a figure is turned about a fixed point • center of rotation: • angle of rotation:

  5. Can you identify the following terms? • rotation: a transformation where a figure is turned about a fixed point • center of rotation: the fixed point (ex. 0,0) • angle of rotation:

  6. Can you identify the following terms? • rotation: a transformation where a figure is turned about a fixed point • center of rotation: the fixed point (ex. 0,0) • angle of rotation: the number of degrees a figure moves clockwise or counterclockwise about the center of rotation

  7. Angle of Rotation -the number of degrees a figure moves clockwise or counterclockwiseabout the center of rotation

  8. Angle of Rotation -the number of degrees a figure moves clockwise or counterclockwiseabout the center of rotation

  9. Identifying Rotations

  10. Identifying Rotations-OYO

  11. 90º Clockwise Rotations Do you see a way to find the image coordinates if you have the original coordinates?

  12. 90º Clockwise Rotations Try: Original Image A (3, -5) B (2, -4) C (4, -1) D (-1, 6) Switch the coordinates and multiply the new y-coordinate by -1. (x,y) (y,-x)

  13. 90º Clockwise Rotations Try: Original Image A (3, -5) A’ (-5, -3) B (2, -4) B’ (-4, -2) C (4, -1) C’ (-1, -4) D (-1, 6) D’ (6, 1) Switch the coordinates and multiply the new y-coordinate by -1. (x,y)  (y,-x)

  14. 90º Counterclockwise Rotations Do you see a way to find the image coordinates if you have the original coordinates?

  15. 90º Counterclockwise Rotations Try: Original Image A (3, -5) B (2, -4) C (4, -1) D (-1, 6) Switch the coordinates and multiply the new x-coordinate by -1. (x,y) (-y,x)

  16. 90º Counterclockwise Rotations Try: Original Image A (3, -5) A’ (5,3) B (2, 4) B’ (-4,2) C (-4, 1) C’ (-1,-4) D (-1, 6) D’ (-6,-1) Switch the coordinates and multiply the new x-coordinate by -1. (x,y) (-y,x)

  17. Rotating a Triangle

  18. Rotating a Triangle

  19. Rotating a Triangle

  20. 180º Rotations To rotate a figure 180º about the origin, multiply each coordinate by -1. (x, y) → (-x, -y)

  21. Rotating a Triangle 180º

  22. Rotating a Triangle 180º

  23. Rotating a Triangle 180º

  24. Rotational Symmetry

  25. A family crest has rotational symmetry for 90º and 180º clockwise (or counterclockwise) rotation. Rotational Symmetry

  26. A family crest has rotational symmetry for 90º and 180º clockwise (or counterclockwise) rotation. Rotational Symmetry

  27. A family crest has rotational symmetry for 90º and 180º clockwise (or counterclockwise) rotation. Rotational Symmetry

  28. A family crest has rotational symmetry for 90º and 180º clockwise (or counterclockwise) rotation. Rotational Symmetry

  29. Dilations -a transformation where a figure stretches or shrinks with respect to a fixed point (center of dilation – ex. 0,0) -in a dilation, a figure and its image are similar

  30. What do you notice about the original and image coordinate points?

  31. What do you notice about the original and image coordinate points? You can multiply the coordinates by a scale factor of 2.

  32. What do you notice about the original and image coordinate points? You can multiply the coordinates by a scale factor of 2. ie. (x,y) (2x,2y)

  33. Or the thing you are multiplying by. If your scale factor is 2, you multiply each coordinate by 2. If it is 0.5, you multiply by 0.5

  34. Dilating a Quadrilateral

  35. Dilating a Quadrilateral

  36. Dilating a Quadrilateral

  37. How do you think we could shrink a dilation?

  38. LET’S TRY

  39. LET’S TRY

  40. LET’S TRY

  41. OYO Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image.

  42. OYO Solution Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image. SOLUTION A’ (-2, 4) B’ (2, 4) C’ (6, 0) D’ (-2, -2)

  43. OYO Solution (cont.) Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image. SOLUTION A’ (-2, 4) B’ (2, 4) C’ (6, 0) D’ (-2, -2)

  44. Finding a Scale Factor To find a scale factor, divide the larger photo coordinates by the smaller photo coordinates (since it was an enlargement). 5/2 = 2.5 2.5/1 = 2.5 12.5/5 = 2.5 2.5/1 = 2.5 The scale factor is 2.5

  45. Try: Find the scale factor. Solution: A’ to A is 3 / 0.5 = 6 and B’ to B is 6/1 = 6. The scale factor is 6. The image (the enlarged line) is 6 times larger than the original line.

  46. Homework Green Challenge: p.744 – 745 # 3 – 5 all, 8 – 13all, 15, 17 *3 graphs needed* p.749 – 750 # 1, 5 – 10 all *4 graphs needed* Blue Challenge: p.744 – 746 # 9 – 19 (odd), 20 – 22 all *5 graphs needed* p.749 – 751 # 1, 5 – 13all, 17, 18 *8 graphs needed*

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