Appendix A — Answer Key, Chapter 3: Symmetry and Transformations

SOL G.RLT.3 (a, b, c, d) · Covers textbook Chapter 3 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 110 across the chapter. Sketch answers describe what a correct drawing shows.

Conventions used in every answer below: the original is the preimage, the result is the image, and image vertices carry a prime. Rotations are counterclockwise about the origin unless stated otherwise. Translation, reflection, and rotation are rigid motions; a dilation multiplies every length by $k$ and leaves every angle measure alone.

The coordinate rules, for reference:

Lesson 3.1 — Lines of Symmetry

Guided practice

1. Square: $4$. Rectangle: $2$.
2. One line — the vertical line through the midpoints of the two parallel sides.
3. None. Folding along a diagonal produces two congruent triangles, but the fold turns one onto the other rather than laying it on top; corresponding points do not land on each other.
4. Regular pentagon: $5$. Regular hexagon: $6$.
5. Three lines join opposite vertices, and three join the midpoints of opposite sides.
6. Stop sign: $8$. Window arch: $1$.

Independent practice

7. a) $3$ b) $1$ c) $0$ d) $8$
8. a) $4$ b) $2$ c) $2$ d) $1$
9. $12$ sides — a regular $n$-gon has $n$ lines of symmetry.
10. $15$ lines. None run vertex to vertex; $15$ is odd, so each line runs from a vertex to the midpoint of the opposite side.
11. A rhombus has two lines of symmetry, and they are its diagonals.
12. A rectangle has two lines of symmetry, and they are the lines through the midpoints of each pair of opposite sides — not its diagonals.
13. $6$. The hexagon supplies six lines, and a pattern that repeats identically in all six sections leaves every one of them intact.
14. One — the vertical line. The bare triangle has three, but a word printed horizontally is not carried onto itself by either of the two slanted folds. (If the word itself is not symmetric, it also breaks the vertical fold; the intended reading is that the word is centered and the sign's outline plus placement leave the vertical fold.)
15. Cutting into congruent halves is not the test. A line of symmetry must fold the figure so that each half lands on top of the other. The diagonal of a parallelogram makes congruent triangles that are rotations of each other, not reflections, so the fold does not match them up.
16. A line of symmetry through a vertex must exit through the point diametrically opposite it. When $n$ is odd there is no vertex there — the opposite position is the middle of a side — so no line joins two vertices.
17. Every line through the center of the circle is a line of symmetry, and there are infinitely many such lines.
18. Sketches vary. Exactly one: an isosceles triangle that is not equilateral, or a kite. Exactly two: a rectangle that is not a square, or a rhombus that is not a square.

Exit ticket 3.1

19. $9$.
20. $2$ — the horizontal and vertical lines through the midpoints of opposite sides.
21. No. No fold carries one half onto the other; the diagonals cut it into congruent triangles but do not reflect one onto the other.
22. A regular $n$-gon has exactly $n$ lines of symmetry.

Lesson 3.2 — Line Symmetry and Point Symmetry

Guided practice

23. The rectangle — two lines of symmetry and point symmetry.
24. The parallelogram.
25. The kite.
26. $A$ lands where $C$ was.
27. Every vertex moved, but each landed exactly where another vertex had been, so the set of vertices — and therefore the outline — is unchanged.
28. No. A regular $n$-gon has point symmetry exactly when $n$ is even, and $5$ is odd.

Independent practice

29. a) both b) both c) line symmetry only d) neither
30. a) both b) line symmetry only ($7$ is odd) c) line symmetry only d) both
31. $12$ lines of symmetry, and it does have point symmetry, because $12$ is even.
32. Its three lines each run from a vertex through the midpoint of the opposite side, so folds work. But a half turn about the center sends each vertex into the middle of the opposite side rather than onto another vertex, so the outline does not land on itself.
33. Any parallelogram that is neither a rectangle nor a rhombus. (Accept also the letter S or Z, or a pinwheel with an even number of blades.)
34. An isosceles trapezoid, a kite, or an isosceles triangle that is not equilateral.
35. Point symmetry — a half turn leaves the design unchanged. It does not guarantee line symmetry; a card could look the same upside down and still have no fold.
36. No line symmetry: a fold would reverse the sweep of the blades. Point symmetry: yes, because four identical blades at $90°$ intervals are carried onto themselves by a $180°$ turn. (A quarter turn also works, which is rotational symmetry of order $4$; point symmetry is the $180°$ case of it.)
37. Counterexample: a parallelogram that is neither a rectangle nor a rhombus. It has point symmetry and no lines of symmetry at all, so one does not follow from the other.
38. Counterexample: a regular pentagon, or any regular polygon with an odd number of sides. Correct rule: a regular $n$-gon has point symmetry exactly when $n$ is even.
39. A half turn moves the vertex in position $k$ to the position $n/2$ steps around the polygon. When $n$ is even, $n/2$ is a whole number of steps and that position holds a vertex, so the figure lands on itself. When $n$ is odd, $n/2$ is not a whole number of steps and the vertex lands mid-side.
40. Reflecting over the first line and then over the second, perpendicular, line is the same as a half turn about their intersection. Both reflections carry the figure onto itself, so the half turn does too — which is point symmetry, with the center at the point where the two lines cross.

Exit ticket 3.2

41. Line symmetry: no. Point symmetry: yes.
42. Yes, because $6$ is even.
43. An isosceles trapezoid (or a kite, or an isosceles triangle that is not equilateral).
44. Line symmetry: some line folds the figure exactly onto itself. Point symmetry: a $180°$ turn about some point carries the figure exactly onto itself.

Lesson 3.3 — Translations and Reflections

Guided practice

45. $A(-4, 1)$, $B(-1, 1)$, $C(-1, 3)$; $A'(1, -3)$, $B'(4, -3)$, $C'(4, -1)$.
46. $(x, y) \rightarrow (x + 5,\ y - 4)$.
47. Every point travels the same distance in the same direction, so all the connecting segments have the same length and the same direction.
48. $x$-axis: $(x, y) \rightarrow (x, -y)$. $y$-axis: $(x, y) \rightarrow (-x, y)$.
49. $y = x$: $(x, y) \rightarrow (y, x)$. $y = -x$: $(x, y) \rightarrow (-y, -x)$.
50. $A = (1, 1)$ lies on the mirror line $y = x$, and a point on the line of reflection does not move.

Independent practice

51. a) $(-2, 3)$ b) $(-9, -3)$ c) $(-6, 9)$
52. $D'(-2, -3)$, $E'(1, -3)$, $F'(1, 1)$. Rule: $(x, y) \rightarrow (x - 3,\ y - 5)$.
53. $x$-axis: $(-5, -2)$. $y$-axis: $(5, 2)$. $y = x$: $(2, -5)$. $y = -x$: $(-2, 5)$.
54. Over the $x$-axis: $J'(2, -3)$, $K'(6, -3)$, $L'(6, -8)$. Over the $y$-axis: $J'(-2, 3)$, $K'(-6, 3)$, $L'(-6, 8)$.
55. $1 - 7 = -6$ and $3 - (-2) = 5$, so $(x, y) \rightarrow (x - 6,\ y + 5)$. The image of $(0, 0)$ is $(-6, 5)$.
56. $(9, 4)$. A reflection undoes itself, so apply the same rule again.
57. $(-3, 5)$. Subtract what was added: $-1 - 2 = -3$ and $-3 + 8 = 5$.
58. Rule $(x, y) \rightarrow (x + 7,\ y + 2)$. After one turn $(10, 6)$; after two turns $(17, 8)$.
59. $(-5, -2)$.
60. Reflecting over $y = x$ swaps the coordinates and does nothing else — it does not change signs. The correct image is $(9, 4)$. The student applied the $y = -x$ rule.
61. A reflection matches each point with a point on the opposite side of the mirror line, which flips the sense in which the vertices are read — counterclockwise becomes clockwise. A translation moves every point the same way, so the vertices keep their order and the sense is unchanged.
62. The point lies on the line of reflection.

Exit ticket 3.3

63. $(3, 1)$.
64. Over the $x$-axis: $(3, 7)$. Over $y = x$: $(-7, 3)$.
65. $-4 - 2 = -6$ and $9 - 2 = 7$, so $(x, y) \rightarrow (x - 6,\ y + 7)$.
66. Reflection over $y = -x$.

Lesson 3.4 — Rotations and Dilations

Guided practice

67. $(x, y) \rightarrow (-y,\ x)$.
68. $180°$: $(x, y) \rightarrow (-x, -y)$. $270°$: $(x, y) \rightarrow (y, -x)$.
69. The $180°$ rotation. It appeared in Lesson 3.2 as the half turn that defines point symmetry.
70. $k = 2$ in the left panel and $k = \tfrac12$ in the right. The left one is the enlargement.
71. That every image vertex lies on the ray from the center through its corresponding preimage vertex, at $k$ times the distance from the center.
72. The dilation. It multiplies every length by $k$; it leaves every angle measure unchanged.

Independent practice

73. a) $(-4, 1)$ b) $(-2, -3)$ c) $(5, 0)$
74. a) $(-6, 1)$ b) $(2, 8)$
75. a) $(3, -5)$ b) $(7, 4)$
76. $P'(-1, 1)$, $Q'(-1, 4)$, $R'(-3, 4)$.
77. $k = \tfrac12$: $(4, -3)$. $k = 3$: $(24, -18)$.
78. $k = 5$ — both $10/2$ and $25/5$ give $5$.
79. $W'(3, 3)$, $X'(12, 3)$, $Y'(12, 9)$, $Z'(3, 9)$. The preimage is $3$ by $2$ with perimeter $10$; the image is $9$ by $6$ with perimeter $30$. The perimeter is multiplied by $k = 3$, the same factor as every length.
80. $(10, -5)$. The angles are unchanged — a dilation preserves angle measure.
81. $90°$ counterclockwise: $(0, 3)$. $90°$ clockwise (the $270°$ counterclockwise rule): $(0, -3)$.
82. The student swapped the coordinates but forgot the negative. The $90°$ rule is $(x, y) \rightarrow (-y, x)$, so the image is $(-2, 5)$.
83. A dilation does not change angle measures at all. It multiplies every length by $3$; every angle keeps its measure, which is precisely why the image is similar to the preimage.
84. A rotation about the origin moves each point along a circle centered at the origin, so its distance from the origin cannot change. Computing that distance before and after is therefore a free check: if $(3, 4)$ is $5$ units out and your image is not, a sign is wrong.
85. A dilation preserves angle measures but multiplies every side length by $k$. Congruent figures need equal side lengths; similar figures need equal angles and proportional sides. So unless $k = 1$, the image is similar but not congruent.

Exit ticket 3.4

86. $(-2, -6)$.
87. $(6, -2)$.
88. $(3, -4)$.
89. The dilation.

Lesson 3.5 — Naming a Transformation from Its Image

Guided practice

90. Case D. The image is visibly larger than the preimage, and only a dilation changes size — no computation needed to rule out the other three.
91. Case B.
92. Case A.
93. Case C is a $180°$ rotation about the origin. Both coordinates changed sign, which a reflection over the origin would also do, but reading $P$, $Q$, $R$ around the preimage and $P'$, $Q'$, $R'$ around the image gives the same direction. A reflection always reverses that direction, so it is a rotation.
94. First a reflection over the $x$-axis, then a translation $5$ units left.
95. Because the second transformation acts on wherever the first one left the figure. Reflecting and then translating puts the figure in a different place from translating and then reflecting, so a description of a composition is incomplete without the order.

Independent practice

96. a) reflection over the $x$-axis b) translation c) dilation from the origin d) rotation $90°$ counterclockwise about the origin
97. Translation rule for (b): $(x, y) \rightarrow (x + 5,\ y + 3)$. Dilation in (c): $k = 2$.
98. A reflection. Same side lengths rules out a dilation; reversed orientation rules out a translation and a rotation, since both preserve orientation.
99. Reflect over the $x$-axis: $(2, -5)$. Then translate $4$ right: $(6, -5)$.
100. Translate first: $(6, 5)$. Then reflect over the $x$-axis: $(6, -5)$. The two orders happen to agree here, because a horizontal translation and a reflection over a horizontal line do not interfere. Reversing the order of a vertical translation and this reflection would not agree — which is why the order must always be stated rather than assumed harmless.
101. Reflect over the $y$-axis: $(-3, -1)$. Then translate $6$ up: $(-3, 5)$.
102. A dilation from the origin with $k = \tfrac32$ ($9/6 = 1.5$ and $12/8 = 1.5$). The angles did not change; only the lengths did.
103. Reversed orientation rules a rotation out. A $180°$ rotation preserves orientation; only a reflection reverses it. The correct name is a reflection.
104. Size is a single comparison that eliminates one of the four possibilities outright and can be judged by eye. Orientation requires reading the vertex order around both figures, which is slower and more error-prone. Doing the cheap test first means you only do the careful one when it can still matter.
105. The identity — reflecting over the same line twice returns every point to where it started, which is equivalent to a translation by $(0, 0)$ or a rotation of $0°$.
106. Answers vary. One correct composition: reflect over the $y$-axis, sending $(1, 1)$ to $(-1, 1)$, then translate $4$ units up, sending it to $(-1, 5)$. Order: reflection first, translation second. (Accept any two named transformations that compose correctly, with the order stated.)

Exit ticket 3.5

107. A dilation, with $k = \tfrac12$.
108. A reflection.
109. A translation, with rule $(x, y) \rightarrow (x - 3,\ y + 2)$.
110. Is it the same size? (No → dilation.) Is the orientation reversed? (Yes → reflection.) Did every point move the same distance in the same direction? (Yes → translation; no → rotation.)

Chapter 3 Review

Review 1 (G.RLT.3 a, b).

The two rules used for the regular polygons: a regular $n$-gon has exactly $n$ lines of symmetry, and it has point symmetry exactly when $n$ is even.

Review 2 (G.RLT.3d). $A(2, 1)$, $B(5, 1)$, $C(5, 5)$.

The first five images are congruent to $\triangle ABC$, because a translation, a reflection, and a rotation are all rigid motions and preserve every distance. The dilation's image is only similar: its angles match, but every side length is twice as long, so it is the same shape at a different size.

Review 3 (G.RLT.3 c, d). $D(1, 2)$, $E(4, 2)$, $F(4, 4)$ and $D'(-1, -2)$, $E'(-4, -2)$, $F'(-4, -4)$.

• $DE = 3$ and $D'E' = 3$; $EF = 2$ and $E'F' = 2$. Corresponding sides are equal, so the image is congruent to the preimage and the transformation is a rigid motion.
• A single transformation: a rotation of $180°$ about the origin. Every point's coordinates both changed sign, which is exactly $(x, y) \rightarrow (-x, -y)$.
• A composition: a reflection over the $x$-axis followed by a reflection over the $y$-axis (either order gives the same result here, because the two mirror lines are perpendicular). Checking $D$: $(1, 2) \rightarrow (1, -2) \rightarrow (-1, -2)$.
• A reflection over any single named line is ruled out by orientation. Reading $D$, $E$, $F$ around the preimage and $D'$, $E'$, $F'$ around the image gives the same rotational direction, and every reflection reverses it. Checking the four rules individually confirms it: the $x$-axis rule gives $(1, -2)$, the $y$-axis rule gives $(-1, 2)$, the $y = x$ rule gives $(2, 1)$, and the $y = -x$ rule gives $(-2, -1)$ — none of which is $D'(-1, -2)$.

Workbook-only items

Page 2, fill in the blanks. A line of symmetry is a line you could fold the figure along so the two halves land exactly on each other.

Page 3, fill in the blank and reference table. A regular $n$-gon has exactly $n$ lines of symmetry. Square $4$; rectangle (not square) $2$; rhombus (not square) $2$; parallelogram (neither) $0$; isosceles trapezoid $1$; kite $1$; equilateral triangle $3$; scalene triangle $0$.

Page 7, fill in the blanks. Line symmetry — some line folds the figure onto itself. Point symmetry — some point is a center about which a $180°$ turn carries the figure onto itself.

Page 8, fill in the blank. A regular $n$-gon has point symmetry exactly when $n$ is even.

Page 11, fill in the blank. $(x, y) \rightarrow (x + a,\ y + b)$.

Page 12, reflection frame. $x$-axis $(x, -y)$, sign of $y$ changes. $y$-axis $(-x, y)$, sign of $x$ changes. $y = x$: $(y, x)$, coordinates swap. $y = -x$: $(-y, -x)$, coordinates swap and both signs change.

Page 14, fill in the blank. To undo a translation, subtract what was added.

Page 16, rotation frame and watch-out. $90°$: $(-y, x)$. $180°$: $(-x, -y)$. $270°$: $(y, -x)$. A $270°$ counterclockwise turn is the same as a $90°$ clockwise turn.

Page 17, dilation blank and comparison table. $(x, y) \rightarrow (kx,\ ky)$.

Page 20, the three questions. (1) Is it the same size? If no → dilation. (2) Is the orientation reversed? If yes → reflection. (3) Did every point move the same distance in the same direction? If yes → translation; if no → rotation.

Pages 5, 9, 13, and 18, blank grids and sketches. Any assigned plot or sketch. Expected conventions: label the preimage with plain letters and the image with primes; mark the center of a rotation or dilation and the mirror line of a reflection; join corresponding vertices with light dashed segments when showing a translation, and draw rays from the center when showing a dilation.