Chapter 2 Rigid Transformations Geometry. For 1-10, determine if the following statements are always, sometimes, or never true.

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Chapter 2 Rigid Transformations Geometry Name For 1-10, determine if the following statements are always, sometimes, or never true. 1. Right triangles have line symmetry. 2. Isosceles triangles have line symmetry.

Every regular polygon has three lines of symmetry. 8. Every sector of a circle has a line of symmetry. 9. Every parallelogram has rotational symmetry. 10.

1 Chapter 2 Rigid Transformations Geometry Name For 1-10, determine if the following statements are always, sometimes, or never true. 1. Right triangles have line symmetry. 2. Isosceles triangles have line symmetry. 3. Every rectangle has line symmetry. 4. Every rectangle has exactly two lines of symmetry. 5. Every parallelogram has line symmetry. 6. Every square has exactly two lines of symmetry. 7. Every regular polygon has three lines of symmetry. 8. Every sector of a circle has a line of symmetry. 9. Every parallelogram has rotational symmetry. 10. A rectangle has 90, 180, and 270 angles of rotation. 11. Draw a quadrilateral that has two pairs of congruent sides and exactly one line of symmetry. 12. Draw a figure with infinitely many lines of symmetry. 13. Draw a figure that has one line of symmetry and no rotational symmetry. 14. Fill in the blank: A regular polygon with n sides has lines of symmetry. For 15-19, find all lines of symmetry for the letters below

For 21-25, determine if the words below have line

2 Do any of the letters above have rotational symmetry? If so, which one(s) and what are the angle(s) of rotation? For 21-25, determine if the words below have line symmetry or rotational symmetry. 21. OHIO 22. MOW 23. WOW 24. KICK 25. Pod For 26-28, trace each figure and then draw in all lines of symmetry. 26.

3 For 29-34, find the angle(s) of rotation for each figure

4 For 35-37, determine if the figures have line symmetry or rotational symmetry. Identify all lines of symmetry and all angles of rotation

37. 38. What is the difference between a vector and a ray? For 39-45, use the translation. 39. What is the image of A(-6, 3)? 40. What is the image of B(4, 8)? 41. What is the preimage of C (5, -3)?

5 What is the difference between a vector and a ray? For 39-45, use the translation. 39. What is the image of A(-6, 3)? 40. What is the image of B(4, 8)? 41. What is the preimage of C (5, -3)? 42. What is the image of A? 43. What is the preimage of D (12, 7)? 44. What is the image of A? 45. Plot A, A, A, and A from the questions above. What do you notice? Write a conjecture. For 46-49, the vertices of ΔABC are A(-6, -7), B(-3, -10), and C(-5, 2). Find the vertices of ΔA B C, given the translation rules below

6 For 50-53, ΔA B C is the image of ΔABC. Write the translation rule

54. Verify that a translation is an isometry using the triangle from #52. 55.

7 54. Verify that a translation is an isometry using the triangle from # If ΔA B C was the preimage and ΔABC was the image, write the translation rule for #53. For 56-58, name each vector and find its component form For 59-61, plot and correctly label each vector

8 62. The coordinates of ΔDEF are D(4, -2), E(7, -4), and F(5, 3). Translate ΔDEF using the vector and find the coordinates of ΔD E F. 63. The coordinates of quadrilateral QUAD are Q(-6, 1), U(-3 7), A(4, -2), and D(1, -8). Translate QUAD using the vector and find the coordinates of Q U A D. For 64-66, write the translation rule as a translation vector For 67-69, write the translation vector as a translation rule Which letter is a reflection over a vertical line of the letter b? 71. Which letter is a reflection over a horizontal line of the letter b? For 72-83, reflect each shape over the given line. 72. y-axis

9 73. x-axis 74. y = x = x-axis

10 77. y-axis 78. y = x 79. y = -x 80. x = 2

11 81. y = y = -x 83. y = x For 84-87, the vertices of ΔABC are A(-5, 1), B(-3, 6), and C(2, 3). 84. Plot ΔABC on the coordinate plane. 85. Reflect ΔABC over y = 1. Find the coordinates of ΔA B C. 86. Reflect ΔA B C over y = -3. Find the coordinates of ΔA B C. 87. What one transformation would be the same as this double reflection?

12 For 88-91, the vertices of ΔDEF are D(6, -2), E(8,-4), and F(3,-7). 88. Plot ΔDEF on the coordinate plane. 89. Reflect ΔDEF over x = 2. Find the coordinates of ΔD E F. 90. Reflect ΔD E F over x = -4. Find the coordinates of ΔD E F. 91. What one transformation would be the same as this double reflection? For 92-95, the vertices of ΔGHI are G(1, 1), H(5, 1), and I(5, 4). 92. Plot ΔGHI on the coordinate plane. 93. Reflect ΔGHI over the x-axis. Find the coordinates of ΔG H I. 94. Reflect ΔG H I over the y-axis. Find the coordinates of ΔG H I. 95. What one transformation would be the same as this double reflection? 96. Describe the relationship between the line of reflection and the segments connecting the preimage and the image points. 97. Repeat the steps from problem #94 with a line of reflection that passes through the triangle. 98. If you rotated the letter p 180 counterclockwise, what letter would you have? 99. If you rotated the letter p 180 clockwise, what letter would you have? Why do you think that is? 100. A 90 clockwise rotation is the same as what counterclockwise rotation? 101. A 270 clockwise rotation is the same as what counterclockwise rotation? 102. Rotating a figure 360 is the same as what other rotation?

13 For , rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin

15 For , find the measure of x in the rotations below

16 114. For , find the angle of rotation for the graphs below. The center of rotation is the origin and the lighter gray figure is the preimage

17 For , the vertices of ΔGHI are G(-2, 2), H(8, 2), and I(6,8) Plot ΔGHI on the coordinate plane Reflect ΔGHI over the x-axis. Find the coordinates of ΔG H I Reflect ΔG H I over the y-axis. Find the coordinates of ΔG H I What one transformation would be the same as this double reflection? 122. Explain why the composition of two or more isometries must also be an isometry What one transformation is equivalent to a reflection over two parallel lines? 124. What one transformation is equivalent to a reflection over two intersecting lines? For , use the graph of the square below Perform a glide reflection over the x-axis and to the right 6 units. Write the new coordinates What is the rule for this glide reflection? 127. What glide reflection would move the image back to the preimage? 128. Start over. Would the coordinates of a glide reflection where you move the square 6 units to the right and then reflect over the x-axis be any different than #125? Why or why not? For , use the graph of the triangle below Perform a glide reflection over the y-axis and down 5 units. Write the new coordinates What is the rule for this glide reflection?

131. What glide reflection would move the image back to the iamge? For 132-136, use the graph of the triangle below. 132. Reflect the preimage over y = -1 followed by = -7. Write the new coordinates.

18 131. What glide reflection would move the image back to the iamge? For , use the graph of the triangle below Reflect the preimage over y = -1 followed by = -7. Write the new coordinates What one transformation is this double reflection the same as? 134. What one transformation would move the image back to the preimage? 135. Start over. Reflect the preimage over y = -7, then y = -1. How is this different than #132? 136. Write the rules for #132 and #135. How do they differ? For , use the graph of the trapezoid below Reflect the preimage over y = -x then the y-axis. Write the new coordinates What one transformation is this double reflection the same as? 139. What one transformation would move the image back to the preimage? 140. Start over. Reflect the preimage over the y-axis, then y = -x. how is this different than #137?

141. Write the rules for #137 and #140. How do they differ? For 142-151, fill in the blanks or answer the questions. 142. Two parallel lines are 7 units apart.

19 141. Write the rules for #137 and #140. How do they differ? For , fill in the blanks or answer the questions Two parallel lines are 7 units apart. If you reflect a figure over both, how far apart will the preimage and image be? 143. After a double reflection over parallel lines, a preimage and its image are 28 units apart. How far apart are the parallel lines? 144. A double reflection over the x and y axes is the same as a of What is the center of rotation for #144? 146. Two liens intersect at an 83 angle. If a figure is reflected over both lines, how far apart will the preimage be? 147. A preimage and its image are 244 apart. If the preimage was reflected over two intersected liens, at what angle did they intersect? 148. A rotation of 45 clockwise is the same as a rotation of counterclockwise After a double reflection over parallel lines, a preimage and its image are 62 units apart. How far apart are the parallel lines? 150. A figure is to the left of x = a. If it is reflected over x = a followed by x = b and b > a, then the preimage and image are units apart and the image is to the of the preimage A figure is to the left of x = a. If it is reflected over x = b followed by x = a and b > a, then the preimage and image are units apart and the iamge is to the of the preimage.

Transformations. Working backwards is performing the inverse operation. + - and x 3. Given coordinate rule

Transformations. Working backwards is performing the inverse operation. + - and x 3. Given coordinate rule Transformations In geometry we use input/output process when we determine how shapes are altered or moved. Geometric objects can be moved in the coordinate plane using a coordinate rule. These rules can