1. Lesson 7.1 Rigid Motion in a Plane Today, we will learn to… > identify the 3 basic transformations > use transformations in real-life situations

2. Transformations The original figure is called the ____________ and the new figure is called the ____________. preimage image

3. Transformations Preimage: A, B, C, D Image: A ’, B ’, C ’, D ’

6. Rotation

7. Translation

8. Isometries preserve length, angle measures, parallel lines, & distances between points

9. Theorems 7.1, 7.2, & 7.4 Reflections, translations, and rotations are isometries.

10. 1. Name and describe the transformation. reflection over  ABC  the y-axis A’B’C’A’B’C’

11. 2. Name the coordinates of the vertices of the preimage and image. (-4,0) (-4,4) (4,0) (4,4) (0,4)

12. 3. Name and describe the transformation. reflection over ABCD  x = -1 HGFE

13. 4. Is the transformation an isometry? Explain. NO YES NO

14. 5. The mapping is a reflection. Which side should have a length of 7? Explain. WX = 7

15. 6. Name the transformation. Find x and y. Reflection x =y = 404

16. Reflection 7. Name the transformation. Find x and y. x =y =124

17. 8. Name the transformation. Find a, b, c, and d. a = b = c = d = 73 53 15 8 reflection

18. 9. Name the transformation. Find p, q, and r. p = q = r = 19 3 7.5 rotation

19. 10. Name the transformation and complete this statement  GHI   ____ LKP reflection

20. 11. Name the transformation that maps the unshaded turtle onto the shaded turtle reflectiontranslation rotation

22. Lesson 7.2 Reflections Today, we will learn to… > identify and use reflections > identify relationships between reflections and line symmetry

23. Reflection 2 images required

24. 1. Is this a reflection?What is the line of reflection? YES x = -2

25. 2. Is this a reflection? NO

26. 3. Is this a reflection?What is the line of reflection? YES y = 1

27. 4. Is this a reflection? What is the line of reflection? YES y = x

28. 5. Is this a reflection? What is the line of reflection? YES y = - x

29. When can I use this in “Real Life?” Finding a minimum distance Telephone Cable - Pole Placement TV cable (Converter Placement) Walking Distances Helps you work smarter not harder

30. Finding a minimum distance 6. A new telephone pole needs to be placed near the road at point C so that the length of telephone cable (AC + CB) is a minimum distance. Two houses are at positions A and B. Where should you locate the telephone pole?

31. A B C A’A’ Finding a minimum distance 1) reflect A 2) connect A ’ and B 3) mark C

32. GSP

33. Line of Symmetry 1 image reflects onto itself

34. 7. How many lines of symmetry does the figure have? 1 2 3 4 5 6 7 8

35. 8. How many lines of symmetry does the figure have? 2

36. m  A = can be used to calculate the angle between the mirrors in a kaleidoscope n = the number of lines of symmetry 180˚ n

37. 1 2 3 4 5 6 7 8 180˚ 8 = 22.5˚

38. http://kaleidoscopeheaven.org 180˚ 9 = 20˚

39. 10. Find the angle needed for the mirrors in this kaleidoscope. 180˚ 4 = 45˚

40. Project? 1) Identify a reflection in a flag 2) Identify a line of symmetry

41. Reflection Line of Symmetry

42. Reflection Line of Symmetry

43. Section 7.2 Practice Sheet !!!

45. Lesson 7.3 Rotations students need tracing paper Today, we will learn to… > identify and use rotations

46. Rotation Angle of Rotation? Center of Rotation? Direction of Rotation?

47. Clockwise rotation of 60° Center of Rotation? Angle of Rotation? 60˚

48. Counter- Clockwise rotation of 40° 40°

49. Theorem 7.3 A reflection followed by a reflection is a rotation. If x˚ is the angle formed by the lines of reflection, then the angle of rotation is 2x°.

50. A B’ A’ A’’ B’’ B 2x˚ x˚

51. 1. What is the degree of the rotation? 140˚ 70˚ A A’A’ A ’’

52. 2. What is the degree of the rotation? 110˚ A A’ A’’ ? 125˚ 55˚

53. 3. Use tracing paper to rotate ABCD 90º counterclockwise about the origin. B (4, 1) Figure ABCD Figure A ' B ' C ' D ' A (2, –2) A ’ (2, 2)B ‘ (–1, 4) C (5, 1) C ‘ (–1,5) D (5, –1) D ‘ (1, 5)

54. Rotational Symmetry A figure has rotational symmetry if it can be mapped onto itself by a rotation of 180˚ or less. I had another dream….

55. 6. Describe the rotations that map the figure onto itself. = 45˚ 360˚ 8 1 2 3 4 5 6 7 8 ___ rotational symmetry 45˚

56. Describe the rotations that map the figure onto itself. 360 2 = 180˚ 1 2 ____ rotational symmetry 180˚

57. Describe the rotations that map the figure onto itself. ___ rotational symmetry no

58. Describe the rotational symmetry. 1 2 3 4 56 360 6 = 60˚ 60˚ rotational symmetry

59. Which segment represents a 90˚clockwise rotation of AB about P? CD

60. Which segment represents a 90˚counterclockwise rotation of HI about Q? LF

61. Project? 1) Identify a rotation in a flag 2) Identify rotational symmetry in a flag

62. Rotation 60° Rotational symmetry

63. A B C D E J P KM HF G L Section 7.3 Practice!!!

65. Lesson 7.4 Translations and Vectors Today, we will learn to… > identify and use translations

66. Translation

67. One reflection after another in two parallel lines creates a translation. mn THEOREM 7.5

68. PP '' is parallel to QQ '' km Q P Q 'Q ' P ' Q '' P '' PP '' is perpendicular to k and m. ______________ _______

69. km Q P Q 'Q ' P ' Q '' P '' 2d2d d The distance between P and P” is 2d, if d is the distance between the parallel lines.

70. Name two segments parallel to YY ” XX ” ZZ ”

71. Find YY ” 6 cm 12 cm XX ” = ZZ ” = 12 cm

72. A translation maps  XYZ onto which triangle? X”Y”Z”X”Y”Z”

73. Name two lines  to XX ” line k line m

74. A translation can be described by coordinate notation. (x, y)  (x + a, y + b) describes movement left or right describes movement up or down (x, y)  (x + 12, y - 20) means to translate the figure… right 12 spaces & down 20 spaces – + + –

75. 1. (x, y)  (x + 1, y – 9) Use words to describe the translation. 2. (x, y)  (x – 2, y + 7) right 1 space, down 9 spaces left 2 spaces, up 7 spaces

76. (x, y)  (x + 5, y – 3)

77. 3. left 5, down 10 Write the coordinate notation described. 4. up 6 (x – 5, y – 10) (x, y + 6) (x, y) 

78. 5. Describe the translation with coordinate notation. -2 +3 -2 (x,y)  (x – 2, y + 3)

79. 6. Describe the translation with coordinate notation. -7 -2 -7 (x,y)  (x – 7, y – 2) -2 -7 -2 -7

80. 7. A triangle has vertices (-4,3); (0, 4); and (3, 2). Find the coordinates of its image after the translation (x, y)  (x + 4, y – 5) (-4, 3)  (-4 + 4, 3 – 5)  (3, 2)  (3 + 4, 2 – 5)  (0, 4)  (0 + 4, 4 – 5)  (7, -3) (4, -1) (0,-2)

81. Graphically, it would be… (x, y)  (x + 4, y – 5) (-4, 3)  (0, -2) (0, 4)  (4, -1) (3, 2)  (7, -3)

82. preimage  image (x, y)  (x + 6, y – 2) 8. Find the image of (-4, 5) 9. Find the preimage of (9, 5) (2, 3) (3, 7) (-4, 5)  (-4 + 6, 5 – 2)  ( __, __ ) ( _, _ )  ( x + 6, y – 2)  ( 9, 5 ) x + 6 = 9 y – 2 = 5

83. A vector is a quantity that has both direction and magnitude (size). A vector can be used to describe a translation.

84. 3 units up 5 units right initial point terminal point BA 5 4 2 B A

85. The vector component form combines the horizontal and vertical components. (x, y)  (x + 5, y + 3) Write this in coordinate notation form

86. C 10. What is the component form of the vector used for this translation?

87. 11. Name the vector and write its component form. XY X Y Write this in coordinate form. (x,y)  (x + 5, y – 3)

88. 12) Describe the translation which maps  ABC onto  A ’ B ’ C ’ by writing the translation in coordinate form and in vector component form. A(3,6); B(1,0); C(4,8); A ’ (1,2); B ’ (-1,-4); C ’ (2,4) (x, y)  (x – 2, y – 4)  – 2, – 4 

89. Project? 1) Identify a translation in a flag

90. Translation

91. Project? Two Objects Required One Object Only ReflectionLine of Symmetry Rotation Rotational Symmetry Translation

93. Lesson 7.5 Glide Reflections and Compositions students need worksheets and tracing paper

94. glide reflection Example #1 Example #2 To be a “glide” reflection, the translation must be parallel to the line of reflection.

95. NOT a glide reflection NOT a glide reflection These are just examples of a translation followed by a reflection.

96. Two or more transformations are combined to create a composition.

97. A A (2, 4) A ’ (, ) A ’’ (, ) 1. translation: (x,y)  (x, y+2) reflection: in the y-axis 26-26 A’A’ A”A”

98. 2. reflection: in y = x translation: (x,y)  (x+2, y-3) A A (-3, -2) A ’ (, ) A ’’ (, ) -2-30-6 A’A’ A”A”

99. A A”A” B A’A’ B’B’ B ’’ A ’’ (-1,- 4) and B ’’ ( 2,- 1) 3. translation: (x,y)  (x-3, y) reflection: in the x-axis A (2, 4) and B (5, 1)

100. 4. translation: (x,y)  (x, y+2) reflection: in y = -x A (0, 4) and B (3, 2). A ’’ (-6, 0) and B ’’ ( -4,-3) B A B’B’ A’A’ B”B” A”A”

101. 5. Describe the composition. Reflection:in x-axis Translation: (x,y)  (x + 6,y + 2)

102. 6. Describe the composition. Reflection:in y = ½ Rotation:90˚ clockwise about (1,-3)

103. Practice How do we get better?