1. Motion Geometry Part I Geometry Solve Problems Organize Model Compute
Communicate
Measure
Reason
Analyze

2. Transformations

3. Transformations
A transformation is a change in position, shape, or size of a figure.

4. Example
Putting together a jigsaw puzzles is an example motion geometry in action and can be used to illustrate transformations.

5. How does it work?
When you get a new jigsaw puzzle, you dump all the pieces out of the box onto a table.

6. What do you do next?
You probably turn the pieces over so that they are all face up.
You might adjust the angle of the pieces.
You might slide a piece across the table.
Each of these represents a transformation of the piece.

7. Each of these translations has a special name.
Flipping the piece over is an example of a reflection (or flip).
Changing the angle of the piece is an example of a rotation (or turn).
Moving the piece across the table is an example of a translation (or slide).

8. Isometry
If a figure and the figure formed by transforming it are congruent, the transformation is called an isometry. If a transformation is an isometry, the size and shape of the figure remains the same and only the position of the figure changes.

9. Fact
In an isometry distance is also preserved. Since the figures before and after the transformation are congruent, the distance between corresponding points does not change.

10. What do you think?
Is flipping a puzzle piece an isometry?

11. Solution Yes. The image and object are congruent.
Shape, size, and distance are preserved.

12. What do you think?
Is turning or rotating a puzzle piece an isometry?

13. Solution Yes. The image and object are congruent.
Shape, size, and distance are preserved.

14. What do you think?
Is sliding a puzzle piece across the table an isometry?

15. Solution Yes. The image and object are congruent.
Shape, size, and distance are preserved.

16. Orientation
The orientation of an object refers to the order of its parts as you move around the object in a clockwise or a counter-clockwise direction.

17. Example: What is the orientation of the giraffe’s nose, ears, and tail starting with the nose and going clockwise?

18. Solution
Nose – Ears – Tail

19. What do you think?
If the giraffe is slid to a new position, does its orientation change?

20. Solution No. It is still nose – ears – tail.

21. What do you think?
If the giraffe is turn or rotated, does its orientation change?

22. No. The orientation of the giraffe does not change
No. The orientation of the giraffe does not change. In both cases the order is nose – ears – tail.

23. What do you think?
If the giraffe is reflected, does its orientation change?

24. Yes the orientation changes in a reflection
Yes the orientation changes in a reflection. Starting at the nose and going clockwise, its orientation is now: nose – tail – ears.

25. Translations

26. Translations
 A translation is a transformation that moves all points of a figure the same distance in the same direction.

27. Translations
In order to translate a figure you need to know two things.
How far will it be translated?
In what direction will it be translated?

28. Fact A translation (or slide) preserves size, shape, distance, and
orientation.

29. Terminology
In a transformation, the given figure is called the preimage and the transformed figure is called the image. Points on the image that correspond to points on the preimage are labeled similarly but with primes. A transformation is said to map a figure onto its image.

30. Try It Choose one of your attribute pieces.
Draw an arrow on your paper.
Place your attribute piece at the end of the arrow. Trace around it.
Use the arrow (vector) to represent the direction and distance, translate your attribute piece. Trace around it.

31. Did you align a vertex or a side at the foot of the arrow?
Foot of arrow
Foot of arrow

32. It is more difficult to translate using a vertex than a side
It is more difficult to translate using a vertex than a side. You can slide the side along the arrow. BUT B A C A' B'
preimage
image C' u

33. Do not rotate as you slide.
If you rotate with the vertex alone it is
difficult not to rotate as well as slide the
figure. u B C A

34. Translating Polygons by Construction
A polygon can be translated by translating its vertices and then connecting these points.
So it is only necessary to know how to translate a point in order to know how to translate a polygon.

35. Example
Translate point A according to the given vector.

36. Plan
Construct a parallelogram with the ends of the arrow (vector) and point A as three of its vertices.
The fourth vertex will be, the required image.

37. Use your compass to measure the length of the vector
Copy this length from point A in the general direction of the arrow.

38. Using your compass measure the distance from the end of the arrow to point A.
Copy this distance from the head of the vector.
The intersection of arcs is the fourth vertex.

39. Try It Draw a line segment on your paper and a vector (arrow) near it.
Transform the segment according to the vector.

40. Solution:

41. Application of a translation

42. Frieze Patterns
A frieze pattern is a pattern that repeats itself along a straight line. The pattern may be mapped onto itself with a translation.
Wallpaper borders are practical applications of frieze patterns.
Frieze patterns can be found around the eaves of some old buildings.

43. Translation with dot paper
Translations on dot paper can be accomplished using the slope of the translation vector.

44. Try It

45. Solution 3 4
image
preimage

46. Mathematical Notation of a Translation
A translation, T, that moves an object h units to the right or left and k units up or down is T(h,k).
This may also be written using the following notation. T: (x, y) (x + h, y + k)
If h is positive, the object moves to the right.
If h is negative, the object moves to the left.
If k is positive, the object moves up and
If k is negative, the object moves down.

47. Try It Where would the point (2, -3) be located
after the translation described by T(-5,7)?

48. Solution The point moves left 5 and up 7 so
The point moves from (2, -3) to (-3, 4) under this translation.

49. Try It
Translate the triangle using T(3, -4).

50. Solution
image

51. Try It Find the preimage if the following image
resulted after the translation T(5, -3).
image

52. Solution The flag was moved 5 units right and 3 units down.
To undo this and return the flag to its original position, each point in the flag must be moved 5 units left and 3 units
up.

53. image
preimage

54. Reflections

55. Reflections
If you look in a mirror you see your reflection. Your image looks like you because it is the same size and shape as you. The distance from your nose to your lips is the same in your reflection. However, when you raise your right hand your image raise its left hand.

56. Properties of Reflections
A reflection preserves
size,
shape, and
distance.
It reverses orientation.

57. Image reflector
An image reflector can be used to find the position and orientation of an object after it has been reflected.

58. Try It Use your image reflector to complete the butterfly.
Place the beveled edge along the line of reflection.
Look through the reflector until you see its image.
Trace the image.

59. The Butterfly
Reflect the butterfly.

60. Solution

61. Exploration
Find a point on the left half of your butterfly and mark that point and its image.
Draw a line segment connecting the two points.
Use your compass to compare the distance of each point to the reflecting line.
Use the corner of a piece of paper and test to see whether or not the line between the points is perpendicular to the line of reflection.
Pick another point on the butterfly and try the four steps above again.
Write a conjecture about a point and its reflection over a reflecting line.

62. A Reflection
 A reflection is a transformation in which each point is mapped onto to its image over a line in such a way that the line is the perpendicular bisector of the line segment connecting the point and its image.

63. Reflection In order to reflect a figure you need only know
The position of the mirror or line of reflection.

64. Fact
A polygon can be reflected by construction by reflecting its vertices and then connecting the points.
So it is only necessary to know how to reflect a point in order to know how to reflect a polygon.

65. Reflect the point by construction.

66. Solution Drop a perpendicular from the point to the line.
Extend the perpendicular beyond the line.
Measure the length of the point to the line and copy that length on the other side of the reflecting line along the perpendicular.

67. Solution

68. Try It
Reflect the triangle over the line of reflection by construction.

69. Solution

70. Try It
Find the line of reflection in the following figure.

71. Solution

72. Application
A kaleidoscope is a device containing stationary mirrors and loose pieces of colored glass.
The glass pieces are reflected many times in the mirrors depending upon the number of mirrors and the angles at which they are placed.
As the kaleidoscope is rotated, the pieces of glass move and an ever changing colorful, symmetric pattern is created.

73. How does it work?

74. How does it work?
Not only are the objects placed between the mirrors reflected, but so are the objects in the virtual mirrors that are created.
The angle between the mirrors is critical so that eventually reflections will coincide. The viewing eye piece is usually circular.

75. How does it work?
If an object is placed between the mirrors, it is reflected by both mirrors.
Original Shape
Mirror
Mirror
Virtual Mirror
Virtual Mirror

76. How does it work?
Second reflection

77. How does it work?
Third reflection

78. How does it work?
Fourth reflection

79. Change the angle Suppose the angle is changed to 360.
How will it look after the reflections are complete?

80. Solution

81. Reflecting with dot paper.
To reflect objects on dot paper when the line of reflection has a slope of 1 or -1:
Find the perpendicular from the preimage to the line of reflection by counting dots along the diagonal from the point to the line of reflection.
The image will be the same distance from the line of reflection on the same diagonal but on the other side of the line of reflection.

82. Dot Paper Reflection
Reflect point A over the given line of reflection by counting dots. A

83. Count the Dots
The line of reflection has a slope of 1. Point A is 5 diagonal units from the line of reflection so is 5 diagonal units on the other side of the line of reflection. A 1 5 4 3 2 A'

84. Try It
Reflect the line segment.

85. Solution
Count the dots along the diagonal. A B A' B'

86. Reflection over the y-axis
Reflect the points over the y-axis A B C D
y - axis
x - axis

87. Reflections over the y-axis
Reflect the points over the y-axis. E F H G
y- axis
x- axis

88. Find Coordinates Complete the chart for both the object and its image.
Point
Coordinates of Preimage
Coordinates of Image A B C D E F G H

89. Coordinates of Preimage
Solution
Point
Coordinates of Preimage
Coordinates of Image A
(-8, 9)
(8, 9) B
(1, 6)
(-1, 6) C
(7, -4)
(-7, -4) D
(-5, -6)
(5, -6) E
(-3, 5)
(3, 5) F
(4, 2)
(-4, 2) G
(3, -7)
(-3, -7) H
(-5, -2)
(5, -2)

90. Try It
Write a conjecture giving the coordinate of the image of point (x, y) reflected over the y - axis.

91. Solution
The x-coordinate of the image is the negative of the x-coordinate of the preimage.
The y-coordinate remains the same.
The coordinates of the image are
(-x, y).

92. Notation of Reflections
The mathematical notation for a reflection is a lower case r with the equation of the line of reflection or a letter indicating an axis as a subscript.
To indicate a reflection over the y-axis either rx=0 or ry is used.

93. Notations of Reflections over the y-axis
rx=0(x, y) = (-x, y)
ry(x, y) = (-x, y)

94. Reflections of the x-axis
Reflect the same points over the x-axis.
Make a conjecture as to the coordinates of (x, y) reflected over the x-axis.

95. Solution The x-coordinate of the image and the preimage are the same.
The y-coordinate of the image is the negative of the y-coordinate of the preimage.
The coordinates of the (x, -y)

96. Notations of Reflections over the x-axis
ry=0(x, y) = (x, -y)
rx(x, y) = (x, -y)

97. Reflections over the line y=x
Reflect the points over the line y=x.
Make a conjecture using your results.

98. Solution The x and y coordinates are interchanged.
Therefore the reflected image of (x, y) is (y, x).

99. Notation of Reflections of the line y=x
ry=x(x,y) = (y, x)

100. Use Patty Paper to Reflect
Draw figure to be reflected.
Draw line of reflection.
Fold paper at the line of reflection.
Copy figure on the other side of the fold.
Unfold.

101. Try It Put a pencil in both hands.
If you are right-handed start with your hand together.
If you are left-handed start with your hands apart.
Write you name with both hands at the same time.

102. Right-handed

103. Left-handed

104. Rotations

105. Rotations
A rotation is a rigid motion just as translations and reflections are.
The figure that is rotated cannot bend or change shape.
The figure and its image are congruent under a rigid motion.
A rotation is an isometry.

106. Rotations moves each point on an object a
A rotation is a transformation that
moves each point on an object a
given angle around a given point.

107. Rotations
Some fixed point in the plane is used as the center of the rotation.
Every point in the figure is turned a given number of degrees about the point.

108. Rotations In order to rotate an object you must know
The center of the rotation.
The angle of the rotation.
The direction of the rotation.

109. Rotation Exploration
In order to visualize a rotation, try this. Trace the following letter F and its “string” on a piece of tracing paper.

110. Rotation Exploration Trace a circle with the radius of the string.
Place the letter F on the circle with the other end of the string at point P, the center of the circle.
Move the F around the circle, keeping the end of the string on point P.
Trace the F in several positions around the circle.

111. Rotation Exploration
Your results will look something like this.

112. Notation for a Rotation
The notation for a rotation is an upper case script R with two subscripts.
The first subscript names the point of rotation
The second subscript indicates the degree of the rotation.
Rp, 90o represents a rotation of 90 degrees about point P.

113. Try It
Perform the rotation RA, a (P) by construction. Label the image. A P 

114. Solution Point A is the center of the rotation.
Point P is the point to be rotated.
Connect A and P with a line segment.
Copy angle a with vertex at A and segment AP as the initial side of the angle.
Copy the angle in the counterclockwise direction as indicated.

115. Solution P' P  A

116. Try It
Rotate RP,b AB by construction.
Label the image. B A β

117. Solution β A B A' B' P

118. Dot Paper Rotations
Rotate a point A -900 (clockwise) around point P. Call it point C.
Rotate a point A 900 (counter-clockwise) around point P. Call it point CC.

119. Try It
Use slopes to rotate the point A . P A

120. Solution The slope of the line segment between A and P is 3/2.
The slope of the segment between C and P and between CC and P must be negative 2/3.
The only difference is the direction of the rotation. Use -2/3 or 2/(-3) to find the points.

121. Solution P C A CC

122. Try It
Rotate the line segment 90 degrees about point P. A B P

123. Solution B' B P A' A

124. Coordinate Geometry Use coordinate geometry to make the
following rotations.
R(0,0), 900 (2, 3)
R(-1, -2), (2, 3)

125. Solution
The first image is at (-3, 2).

126. Solution
The second image is at (4, -5).

127. The End