Geometric Transformation: Translation

2. Transformation In real life, we can see the change of position of various objects from one location to another. For example, moving a cabinet to another place, using the lift from the first floor to the fifth floor, seeing a reflection of ourselves in a mirror, a reflection of trees in a clean reservoir, the rotation of a dartboard or a wind turbine, or even playing checkers. All the above examples involve geometric transformation. Transformation is an operation in which an object changes in position and/or shape and/or size. We can say that a transformation pairs up between each point of the pre-image and the corresponding point. In this unit, we study only the transformation that changes positions, but with the same shape and characteristics. This comprises of: • Translation • Reflection • Rotationn

3. Translation In real life, we use many facilities involving translation such as escalators, elevators, tractor wheels, hoist cement machineries, and many others. Consider the translation of points A, B, and C in various directions and distances. A B CA’ B’ P Q C’ Point A’ is obtained by translating point A 3 units horizontally to the right Point B’ is obtained by translating point B 2 units vertically upward Point C’ is obtained by translating point C 3 2 units along the ray PQ

4. A B C A1 B1 C1 A2 B2 C2

5. A B C A1 B1 C1 A2 B2 C2

6. A B C A1 B1 C1 A2 B2 C2

7. A B C A1 B1 C1 A2 B2 C2

8. A B C A1 B1 C1 A2 B2 C2

9. A B C A1 B1 C1 A2 B2 C2

10. A B C A1 B1 C1 A2 B2 C2

11. A B C A1 B1 C1 A2 B2 C2

12. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right.

21. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right. Point B1is 8 units away from point B horizontally to the right.

30. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right. Point B1is 8 units away from point B horizontally to the right. Point C1is 8 units away from point C horizontally to the right. Triangle A1B1C1 is obtained by translating triangle ABC 8 units horizontally to the right.

39. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right. Point B1is 8 units away from point B horizontally to the right. Point C1is 8 units away from point C horizontally to the right. Triangle A1B1C1 is obtained by translating triangle ABC 8 units horizontally to the right. Point A2is 8 units away from point A1vertically above .

48. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right. Point B1is 8 units away from point B horizontally to the right. Point C1is 8 units away from point C horizontally to the right. Triangle A1B1C1 is obtained by translating triangle ABC 8 units horizontally to the right. Point A2is 8 units away from point A1vertically above . Point B2is 8 units away from point B1vertically above .

57. A B C A1 B1 C1 A2 B2 C2 Point A1is 8 units away from point A horizontally to the right. Point B1is 8 units away from point B horizontally to the right. Point C1is 8 units away from point C horizontally to the right. Triangle A1B1C1 is obtained by translating triangle ABC 8 units horizontally to the right. Point A2is 8 units away from point A1vertically above . Point B2is 8 units away from point B1vertically above . Point C2is 8 units away from point C1vertically above . Triangle A2B2C2 is obtained by translating triangle A1B1C1 8 units vertically upward. Triangle A2B2C2 is obtained by translating triangle ABC 8 units horizontally to the right and 8 units vertically upward.

58. Let rectangle ABCD be the pre-image. Construct an image obtained by translating rectangle ABCD parallel to ray MN with the distance the same as the length of the line segment MN A B D C M N

66. Let rectangle ABCD be the pre-image. Construct an image obtained by translating rectangle ABCD parallel to ray MN with the distance the same as the length of the line segment MBN A B D C M N

72. Let rectangle ABCD be the pre-image. Construct an image obtained by translating rectangle ABCD parallel to ray MN with the distance the same as the length of the line segment MN A B D C M NA1 B1 C1D1

73. Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B Y Y’

79. Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B Y’ 6 units down Y

83. Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down Y 4 units right Y’ P QR

84. Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down Y 4 units right Y’ QR P

85. P Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down Y 4 units right Y’ QR

89. P Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down 4 units right Y’ QR Y

93. Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down Y 4 units right Y’ QR P’ Q’R’ P

94. P Triangle A is translated to Triangle A’. Construct an image obtained by translating triangle B so that the distance and direction are the same as triangles A and A’. A′ A B 6 units down Y 4 units right Y’ QR P’ Q’R’ B′

96. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 1 B −4,2 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up.

97. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 B −4,2 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A 2,3 1

98. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 B −4,2 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. 1

99. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 B −4,2 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. 1

100. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 B −4,2 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A′ 6,3 1

101. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A′ 6,3 B −4,2 1

104. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A′ 6,3 B −4,2 B′ 0,2 1

108. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A 2,3 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A′ 6,3 B −4,2 B′ 0,2 C′ −4,1 1

112. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 C −4, −3 D 4, −3 Find the coordinates of points obtained by translating point A 2,3 and B −4,2 parallel to the 𝑥 − axis 4 units to the right and the coordinates of the points obtained by translating points C −4, −3 and D 4, −3 parallel to 𝑦 − axis 4 units up. A′ 6,3 B′ 0,2 C′ −4,1 D′ 4,1 A 2,3 B −4,2 1

113. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A −6,1 B −2,1 C −1,3D −5,3 Let reactangle ABCD with the coordinate A −6,1 , B −4,2 , C −1,3 and D −5,3 be the pre-image. Construct an image obtained by translating rectangle ABCD: 1) Parallel to 𝑥 − axis 7 units to the right 2) Parallel to 𝑦 − axis 4 units down 1

120. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A −6,1 B −2,1 C −1,3D −5,3 A′ 1 B′ 5,1 C′ 6,3D′ 2,3 Let reactangle ABCD with the coordinate A −6,1 , B −4,2 , C −1,3 and D −5,3 be the pre-image. Construct an image obtained by translating rectangle ABCD: 1) Parallel to 𝑥 − axis 7 units to the right 2) Parallel to 𝑦 − axis 4 units down 1

127. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A −6,1 B −2,1 C −1,3D −5,3 A′ 1,1 B′ 5,1 C′ 6,3D′ 2,3 Let reactangle ABCD with the coordinate A −6,1 , B −4,2 , C −1,3 and D −5,3 be the pre-image. Construct an image obtained by translating rectangle ABCD: 1) Parallel to 𝑥 − axis 7 units to the right 2) Parallel to 𝑦 − axis 4 units down 1

131. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 A −6,1 B −2,1 C −1,3D −5,3 A′ 1,1 B′ 5,1 C′ 6,3D′ 2,3 A′ −6, −3 B′ −2, −3 C′ −1, −1D′ −5, −1 Let reactangle ABCD with the coordinate A −6,1 , B −4,2 , C −1,3 and D −5,3 be the pre-image. Construct an image obtained by translating rectangle ABCD: 1) Parallel to 𝑥 − axis 7 units to the right 2) Parallel to 𝑦 − axis 4 units down 1

135. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 Let reactangle ABCD with the coordinate A −6,1 , B −4,2 , C −1,3 and D −5,3 be the pre-image. Construct an image obtained by translating rectangle ABCD: 1) Parallel to 𝑥 − axis 7 units to the right 2) Parallel to 𝑦 − axis 4 units down A −6,1 B −2,1 C −1,3D −5,3 A′ 1,1 B′ 5,1 C′ 6,3D′ 2,3 A′ −6, −3 B′ −2, −3 C′ −1, −1D′ −5, −1 1

136. 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 8 9 10 11-9 -8 -7 -6 -5 -4 -3 -2 -1 𝑥 𝑦 0 Explain how to match point A, the pre-image, to point B, the image being translated for each of the following. 1. A1 1,7 and B1 7,2 2) A2 6, −2 and B2 5,4 A1 1,7 B1 7,2 A2 6, −2 B2 5,4 5 units down 6 units right 6 units right 5 units down 6 units up 1 unit left 1 unit left 6 units up 1

Geometric Transformation: Translation

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Geometric Transformation: Translation