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’

4. R R
REFLECTION

5. R
Rotation

6. R R
Translation

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

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

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

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

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

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

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

14. 6. Name the transformation.
Find x and y.
Reflection
x = 40
y = 4

15. 7. Name the transformation.
Find x and y.
Reflection
x = 12
y = 4

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

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

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

19. reflection translation rotation
11. Name the transformation that maps the unshaded turtle onto the shaded turtle
reflection
translation
rotation

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

22. Reflection
2 images required

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

24. 2. Is this a reflection? NO

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

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

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

28. 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

29. Finding a minimum distance 6. A new telephone pole needs to be
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?

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

31. GSP

32. 1 image reflects onto itself
Line of Symmetry
1 image reflects onto itself

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

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

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

36. 1 2 8 3 7 4 6 5
180˚ 8 =
22.5˚

37. 180˚ 9 =
20˚

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

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

40. Reflection
Line of Symmetry

41. Reflection
Line of Symmetry

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

44. Rotation
Direction of Rotation?
Center of Rotation?
Angle of Rotation?

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

46. Counter-Clockwise rotation of 40°

47. 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°.

48. A B’ A’
2x˚ B x˚
A’’
B’’

49. 1. What is the degree of the rotation?
70˚
140˚

50. 2. What is the degree of the rotation?
55˚
110˚
125˚

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

52. 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….

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

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

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

56. Describe the rotational symmetry.
360 6 = 1
60˚ 2 6 3 5 4
60˚ rotational symmetry

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

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

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

60. 60° Rotational symmetry
Rotation

61. Section 7.3 Practice!!! A B C D E J P K M H F G L

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

64. Translation

65. THEOREM 7.5
One reflection after another in two parallel lines creates a translation. m n R R R

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

67. d
Q '
P ' Q P k m
Q ''
P '' 2d
The distance between P and P” is 2d, if d is the distance between the parallel lines.

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

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

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

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

72. (x, y)  (x + 12, y - 20) means to translate the figure…
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

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

74. (x, y)  (x + 5, y – 3)

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

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

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

78. (-4, 3)  (-4 + 4, 3 – 5)  (0,-2) (4, -1) (0, 4)  (0 + 4, 4 – 5) 
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) 
(0,-2)
(4, -1)
(0, 4)  (0 + 4, 4 – 5) 
(3, 2)  (3 + 4, 2 – 5) 
(7, -3)

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

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

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

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

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

84. 10. What is the component form of the vector used for this translation?4 A D 2 A' D' -5 B 5 C -2 B' C' -4

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

86. 12). Describe the translation which. maps ABC onto A’B’C’ by
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

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

88. Translation
Burundi

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

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

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

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

94. 1. translation: (x,y) (x, y+2) reflection: in the y-axis
A (2, 4) A’ ( , ) A’’ ( , ) 2 6 -2 6

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

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

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

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

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

100. How do we
get better?
Practice

102. Go to the presentation flag_project.