Grade 8 Workbook — Chapter 13: Translations in the Coordinate Plane

SOL 8.MG.3 (a, d) · Companion to Textbook Chapter 13

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 88.

FIGURE: fig10-blank-translation-grids.png supplies four empty grids from $-8$ to $8$ in both directions. Reuse it on any page that needs sketching space; each page below says how many grids it needs.

PAGE 1 — Chapter opener

Chapter 13 · Translations in the Coordinate Plane

Standard 8.MG.3 (a, d)

In this chapter you will:

• Name the preimage and the image, and write image points with prime marks
• Read a rule such as $(x, y) \to (x + 4, y - 3)$ and say it in words
• Turn left, right, up, and down into the right signs
• Find the coordinates of a translated polygon (8.MG.3a)
• Sketch the image of a translated polygon (8.MG.3d)
• Explain why the image is congruent to the preimage

Words to know: transformation · translation · preimage · image · prime notation · translation rule · horizontal translation · vertical translation · congruent · orientation

Every translation in this chapter is vertical, horizontal, or a combination of both. Every coordinate is an integer. Reflections come in Chapter 14.

Drawing convention on every page: the preimage is dashed and the image is solid.

PAGE 2 — What a translation does

13.1 What a Translation Does

FIGURE: fig1-a-slide-with-arrows.png (full width)

Fill in the blanks.

The figure you start with is the $\underline{\hspace{3cm}}$. The figure you end with is the $\underline{\hspace{3cm}}$.

The image of point $A$ is written $\underline{\hspace{2cm}}$ and is read "$A$ $\underline{\hspace{2cm}}$."

In the figure above, every vertex slid $\underline{\hspace{1.5cm}}$ units right and $\underline{\hspace{1.5cm}}$ units up.

All three arrows have the same $\underline{\hspace{3cm}}$ and point in the same $\underline{\hspace{3cm}}$.

1. In "$\triangle ABC$ is translated to $\triangle A'B'C'$," the preimage is $\underline{\hspace{3cm}}$ and the image is $\underline{\hspace{3cm}}$.

Signs and directions. Complete the table.

2. Rule for $6$ units right: $\underline{\hspace{5cm}}$

3. Rule for $5$ units down: $\underline{\hspace{5cm}}$

PAGE 3 — Rules into words, words into rules

Four Directions, Four Rules

FIGURE: fig2-four-directions-four-rules.png (full width)

4. $(x, y) \to (x - 3, y + 7)$ in words: $\underline{\hspace{7cm}}$

5. $(x, y) \to (x + 2, y - 9)$ in words: $\underline{\hspace{7cm}}$

6. True or false: a translation makes the image larger than the preimage. $\underline{\hspace{2cm}}$

   Why? _______________________________________________

7. Write the rule for each translation.

8. Describe each rule in words.

9. For each rule in item 8, circle one: horizontal / vertical / combination

   a) H V C b) H V C c) H V C d) H V C

PAGE 4 — What survives a slide

Congruent, and Facing the Same Way

FIGURE: fig3-congruent-and-same-orientation.png (full width)

Check the boxes. Under a translation, which of these change?

☐ side lengths ☐ angle measures ☐ size ☐ shape ☐ orientation ☐ position

10. Quadrilateral $PQRS$ is translated $10$ units up, with $PQ = 6$ units and $m\angle R = 115^\circ$.

    $P'Q' = \underline{\hspace{2cm}}$ $m\angle R' = \underline{\hspace{2cm}}$

    How do you know without any arithmetic? _______________________________________________

11. Use the figure on page 2. Rule: $\underline{\hspace{5cm}}$

    Vertex $C$ traveled $\underline{\hspace{1.5cm}}$ units $\underline{\hspace{2cm}}$ and $\underline{\hspace{1.5cm}}$ units $\underline{\hspace{2cm}}$.

12. Explain. Why must every vertex move the same distance in the same direction?

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    What would happen to the figure if one vertex moved farther than the others?

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13. Apply it. A game piece sits at $(3, 2)$ and a move slides it $4$ squares right and $1$ square up.

    Rule: $\underline{\hspace{5cm}}$ New location: $\underline{\hspace{3cm}}$

14. Find the error. A student writes "$3$ units left" as $(x, y) \to (x + 3, y)$.

    What went wrong? _______________________________________________

    Correct rule: $\underline{\hspace{5cm}}$

PAGE 5 — Exit ticket 13.1

Exit Ticket 13.1

15. Rule for $5$ units right and $8$ units down: $\underline{\hspace{5cm}}$

16. $(x, y) \to (x - 6, y - 2)$ in words: $\underline{\hspace{6cm}}$

17. Two things a translation preserves: $\underline{\hspace{3cm}}$ and $\underline{\hspace{3cm}}$

    One thing it changes: $\underline{\hspace{3cm}}$

18. Explain the difference between a preimage and an image, and why one is written with a prime mark.

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PAGE 6 — One vertex at a time

13.2 Coordinates of a Translated Polygon

FIGURE: fig4-vertex-by-vertex-count.png (full width)

The method. Apply the rule to $\underline{\hspace{4cm}}$ vertex of the preimage, one at a time.

Worked frame. $\triangle JKL$ under $(x, y) \to (x + 5, y - 4)$:

$$J(-4, 2) \to (-4 + 5,\; 2 - 4) = (\underline{\hspace{1cm}}, \underline{\hspace{1cm}})$$

$$K(-1, 2) \to (\underline{\hspace{1cm}}, \underline{\hspace{1cm}}) \qquad L(-3, 5) \to (\underline{\hspace{1cm}}, \underline{\hspace{1cm}})$$

19. $A(2, 5)$ under $(x, y) \to (x + 3, y)$: $\underline{\hspace{3cm}}$

20. $B(-1, 3)$ under $(x, y) \to (x, y - 4)$: $\underline{\hspace{3cm}}$

21. $C(-4, 1)$ under $(x, y) \to (x + 6, y - 2)$: $\underline{\hspace{3cm}}$

22. $\triangle JKL$ with $J(-4, 2)$, $K(-1, 2)$, $L(-3, 5)$ under $(x, y) \to (x + 5, y - 4)$:

    $J' = \underline{\hspace{2.5cm}}$ $K' = \underline{\hspace{2.5cm}}$ $L' = \underline{\hspace{2.5cm}}$

PAGE 7 — Crossing quadrants, and working backwards

Across the Axes

FIGURE: fig5-combination-across-quadrants.png (full width)

23. $WXYZ$ with $W(2, 1)$, $X(5, 1)$, $Y(5, 4)$, $Z(2, 3)$ under $(x, y) \to (x - 7, y + 2)$:

    $W' = \underline{\hspace{2.5cm}}$ $X' = \underline{\hspace{2.5cm}}$ $Y' = \underline{\hspace{2.5cm}}$ $Z' = \underline{\hspace{2.5cm}}$

24. $\triangle ABC$ with $A(0, 0)$, $B(3, 0)$, $C(0, 4)$, translated $5$ units left and $5$ units down.

    Rule: $\underline{\hspace{5cm}}$

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

Working backwards. Horizontal part $= x' - x$. Vertical part $= y' - y$.

25. $A(6, -3) \to A'(1, -3)$. Rule: $\underline{\hspace{5cm}}$

26. $P(-2, 4) \to P'(-2, -1)$. Rule: $\underline{\hspace{5cm}}$

PAGE 8 — Independent practice, 13.2

Applying the Rule

27. Find the image of each point under $(x, y) \to (x - 4, y + 6)$.

28. $\triangle DEF$: $D(1, -2)$, $E(5, -2)$, $F(4, -5)$ under $(x, y) \to (x - 6, y + 7)$.

    $D' = \underline{\hspace{2.5cm}}$ $E' = \underline{\hspace{2.5cm}}$ $F' = \underline{\hspace{2.5cm}}$

29. $QRST$: $Q(-6, -1)$, $R(-2, -1)$, $S(-2, 2)$, $T(-5, 2)$, translated $8$ units right.

    Rule: $\underline{\hspace{4cm}}$

    $Q' = \underline{\hspace{2.5cm}}$ $R' = \underline{\hspace{2.5cm}}$ $S' = \underline{\hspace{2.5cm}}$ $T' = \underline{\hspace{2.5cm}}$

30. Pentagon $ABCDE$: $A(-3, 3)$, $B(-1, 3)$, $C(0, 5)$, $D(-2, 7)$, $E(-4, 5)$ under $(x, y) \to (x + 4, y - 8)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

    $D' = \underline{\hspace{2.5cm}}$ $E' = \underline{\hspace{2.5cm}}$

31. $\triangle ABC$: $A(2, 3)$, $B(6, 3)$, $C(4, 7)$; image $A'(-1, -2)$, $B'(3, -2)$, $C'(1, 2)$.

    Rule: $\underline{\hspace{5cm}}$ In words: $\underline{\hspace{5cm}}$

32. $M(4, -7) \to M'(4, -2)$. Rule: $\underline{\hspace{4cm}}$ Image of $N(-1, 3)$: $\underline{\hspace{3cm}}$

    Why was one pair of points enough? _______________________________________________

PAGE 9 — Reasoning and applications, 13.2

Think It Through

33. Explain. A preimage vertex sits at the origin and its image also sits at the origin. What translation was applied?

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    Why can a translation not hold one point still while moving the others?

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34. Explain. Why is the $y$-coordinate of every vertex unchanged by a horizontal translation? Use the rule, not a picture.

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35. Apply it. A park is drawn on a grid: swing $(-3, 2)$, slide $(1, 5)$, bench $(-5, -1)$. The park is rebuilt $6$ units east and $4$ units south.

    Rule: $\underline{\hspace{5cm}}$

36. Find the error. A student applies $(x, y) \to (x + 2, y - 3)$ to $(-5, 4)$ and writes $(-7, 7)$.

    Error in the $x$-coordinate: _______________________________________________

    Error in the $y$-coordinate: _______________________________________________

    Correct image: $\underline{\hspace{3cm}}$

PAGE 10 — Exit ticket 13.2

Exit Ticket 13.2

37. Image of $(4, -6)$ under $(x, y) \to (x - 9, y + 1)$: $\underline{\hspace{3cm}}$

38. $\triangle ABC$: $A(-2, -3)$, $B(1, -3)$, $C(-2, 0)$ under $(x, y) \to (x + 5, y + 2)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

39. $R(7, 2) \to R'(2, -4)$. Rule: $\underline{\hspace{5cm}}$

40. The "same trip check": what do you compute for each vertex, and what must be true of the results?

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PAGE 11 — Two steps to a sketch

13.3 Sketching a Translated Image

FIGURE: fig6-sketching-in-two-steps.png (full width)

Step 1. Apply the rule to every vertex and $\underline{\hspace{4cm}}$ the image vertices, labeling each with a $\underline{\hspace{2cm}}$.

Step 2. Connect the image vertices in the $\underline{\hspace{4cm}}$ order as the preimage.

Counting instead of computing. For $(x, y) \to (x + 5, y - 4)$, from each vertex count $\underline{\hspace{1.5cm}}$ squares right and $\underline{\hspace{1.5cm}}$ squares down.

Three checks on a finished sketch.

• The image has the same number of $\underline{\hspace{3cm}}$ as the preimage.
• A horizontal or vertical side has the same $\underline{\hspace{3cm}}$ in both figures.
• The segments joining each vertex to its image are all the same $\underline{\hspace{3cm}}$ and point the same $\underline{\hspace{3cm}}$.

PAGE 12 — Guided sketching

Sketch and List

FIGURE: fig8-practice-translation-set.png (full width — grids a, b, c; sketch directly on them)

41. Grid a), $\triangle ABC$ under $(x, y) \to (x - 6, y + 1)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

42. Grid b), quadrilateral $DEFG$ under $(x, y) \to (x + 7, y - 6)$.

    $D' = \underline{\hspace{2.5cm}}$ $E' = \underline{\hspace{2.5cm}}$ $F' = \underline{\hspace{2.5cm}}$ $G' = \underline{\hspace{2.5cm}}$

43. Grid c), trapezoid $HJKL$, translated $6$ units up.

    Rule: $\underline{\hspace{4cm}}$

    $H' = \underline{\hspace{2.5cm}}$ $J' = \underline{\hspace{2.5cm}}$ $K' = \underline{\hspace{2.5cm}}$ $L' = \underline{\hspace{2.5cm}}$

44. After plotting every image vertex, the next step is $\underline{\hspace{5cm}}$.

    Why does the order matter? _______________________________________________

PAGE 13 — More guided sketching

Plot, Connect, Check

FIGURE: fig10-blank-translation-grids.png (full width — use grids a and b)

45. Grid a). $\triangle PQR$: $P(0, 0)$, $Q(4, 0)$, $R(0, 3)$, translated $5$ units left. Draw the preimage dashed and the image solid.

    Rule: $\underline{\hspace{4cm}}$

    $P' = \underline{\hspace{2.5cm}}$ $Q' = \underline{\hspace{2.5cm}}$ $R' = \underline{\hspace{2.5cm}}$

46. Grid b). Square $(1, 1)$, $(3, 1)$, $(3, 3)$, $(1, 3)$, translated $4$ units down.

    Rule: $\underline{\hspace{4cm}}$ Image corners: $\underline{\hspace{7cm}}$

FIGURE: fig7-finding-the-rule.png (half width)

47. Rule that maps $\triangle RST$ to $\triangle R'S'T'$: $\underline{\hspace{5cm}}$

    Two vertices used to confirm it: $\underline{\hspace{1.5cm}}$ and $\underline{\hspace{1.5cm}}$

48. The three checks, and the error each one catches.

PAGE 14 — Independent sketching

Independent Practice 13.3

FIGURE: fig10-blank-translation-grids.png (full width — grids a, b, c for item 49; grid d for item 50)

49. $\triangle ABC$: $A(-4, 1)$, $B(-1, 1)$, $C(-3, 4)$. Sketch each image and list the coordinates.

50. Quadrilateral $ABCD$: $A(2, -1)$, $B(6, -1)$, $C(5, -4)$, $D(3, -4)$ under $(x, y) \to (x - 8, y + 5)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$ $D' = \underline{\hspace{2.5cm}}$

PAGE 15 — Independent sketching, continued

More Sketching

FIGURE: fig10-blank-translation-grids.png (full width — one grid per item)

51. Pentagon $PQRST$: $P(1, 2)$, $Q(4, 2)$, $R(5, 4)$, $S(3, 6)$, $T(1, 4)$ under $(x, y) \to (x - 6, y - 7)$.

    $P' = \underline{\hspace{2.2cm}}$ $Q' = \underline{\hspace{2.2cm}}$ $R' = \underline{\hspace{2.2cm}}$ $S' = \underline{\hspace{2.2cm}}$ $T' = \underline{\hspace{2.2cm}}$

52. $\triangle ABC$: $A(-2, 2)$, $B(-2, 5)$, $C(-6, 2)$ under $(x, y) \to (x + 8, y - 7)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

53. Quadrilateral $(-1, -1)$, $(2, -3)$, $(4, 0)$, $(1, 2)$, translated $2$ units left and $5$ units up.

    Rule: $\underline{\hspace{4cm}}$ Image: $\underline{\hspace{7cm}}$

54. Explain. How does your finished picture show that the image is congruent to the preimage? Name one measurement you compared.

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55. Explain. Which step guarantees that the orientation is preserved, and what goes wrong if it is done carelessly?

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PAGE 16 — Applications and error hunt, 13.3

Slides in the Real World

FIGURE: fig10-blank-translation-grids.png (full width — grids a and b)

56. Apply it. A quilt block is the square $(0, 0)$, $(4, 0)$, $(4, 4)$, $(0, 4)$, repeated $5$ units right.

    Rule: $\underline{\hspace{4cm}}$ New corners: $\underline{\hspace{7cm}}$

57. Apply it. A stage riser has corners $(2, 1)$, $(6, 1)$, $(6, 3)$, $(2, 3)$ and is moved $3$ units left and $2$ units down.

    Rule: $\underline{\hspace{4cm}}$ New corners: $\underline{\hspace{7cm}}$

58. Find the error. A student sketches an image under $(x, y) \to (x + 3, y - 2)$ by moving only vertex $A$ and connecting $A'$ to the original $B$ and $C$.

    Why can the drawing not be the image? _______________________________________________

    Correct procedure: _______________________________________________

PAGE 17 — Exit ticket 13.3

Exit Ticket 13.3

FIGURE: fig10-blank-translation-grids.png (half width — grids a and b)

59. Grid a). $\triangle ABC$: $A(3, 1)$, $B(6, 1)$, $C(6, 5)$ under $(x, y) \to (x - 9, y - 3)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

60. Grid b). Square $(-5, 1)$, $(-2, 1)$, $(-2, 4)$, $(-5, 4)$, translated $7$ units right and $6$ units down.

    Rule: $\underline{\hspace{4cm}}$ Image corners: $\underline{\hspace{7cm}}$

61. While sketching, you find $B(-3, 5) \to B'(4, 1)$. Rule: $\underline{\hspace{5cm}}$

62. Describe the two-step sketching method in your own words, in two sentences.

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PAGE 18 — Review Part A

Chapter 13 Review · Part A

Identifying the coordinates of a translated image (8.MG.3a)

63. Under $(x, y) \to (x + 7, y - 3)$:

    a) $(0, 0) \to \underline{\hspace{2.5cm}}$ b) $(-5, 2) \to \underline{\hspace{2.5cm}}$ c) $(4, -2) \to \underline{\hspace{2.5cm}}$

64. $\triangle ABC$: $A(-6, 4)$, $B(-3, 4)$, $C(-5, 7)$ under $(x, y) \to (x + 9, y - 9)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$

65. $KLMN$: $K(1, 1)$, $L(5, 2)$, $M(4, 5)$, $N(0, 4)$ under $(x, y) \to (x - 6, y - 3)$.

    $K' = \underline{\hspace{2.5cm}}$ $L' = \underline{\hspace{2.5cm}}$ $M' = \underline{\hspace{2.5cm}}$ $N' = \underline{\hspace{2.5cm}}$

66. $(3, -4)$ translated $10$ units left: $\underline{\hspace{3cm}}$

67. $B(-4, -6) \to B'(2, 1)$. Rule: $\underline{\hspace{4cm}}$ In words: $\underline{\hspace{4cm}}$

68. $A(1, 2) \to A'(1, 2)$. What translation? $\underline{\hspace{4cm}}$

    Explain: _______________________________________________

69. Under $(x, y) \to (x - 4, y + 2)$ a vertex has image $A'(3, -5)$.

    Preimage vertex: $\underline{\hspace{3cm}}$ Forward check: $\underline{\hspace{5cm}}$

70. Pentagon $VWXYZ$: $V(-1, -2)$, $W(2, -2)$, $X(3, 0)$, $Y(1, 2)$, $Z(-2, 0)$ under $(x, y) \to (x + 3, y + 4)$.

    $V' = \underline{\hspace{2.2cm}}$ $W' = \underline{\hspace{2.2cm}}$ $X' = \underline{\hspace{2.2cm}}$ $Y' = \underline{\hspace{2.2cm}}$ $Z' = \underline{\hspace{2.2cm}}$

71. $ABCD$ is translated. The image vertex corresponding to $C$ is $\underline{\hspace{1.5cm}}$.

    If $BC = 9$ units then $B'C' = \underline{\hspace{1.5cm}}$, because $\underline{\hspace{5cm}}$.

72. Explain. Why must integer preimage coordinates and integer translation amounts give integer image coordinates?

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PAGE 19 — Review Part B

Chapter 13 Review · Part B

Sketching a translated image (8.MG.3d)

FIGURE: fig10-blank-translation-grids.png (full width — one grid per item, repeat the figure as needed)

73. $\triangle ABC$: $A(2, 2)$, $B(5, 2)$, $C(5, 6)$ under $(x, y) \to (x - 8, y - 1)$.

    Image: $\underline{\hspace{7cm}}$

FIGURE: fig9-review-translation-set.png (full width — grid b for item 74)

74. Grid b), quadrilateral $ABCD$ under $(x, y) \to (x - 7, y - 6)$.

    $A' = \underline{\hspace{2.5cm}}$ $B' = \underline{\hspace{2.5cm}}$ $C' = \underline{\hspace{2.5cm}}$ $D' = \underline{\hspace{2.5cm}}$

75. Trapezoid $(-6, -2)$, $(-2, -2)$, $(-3, -5)$, $(-5, -5)$ under $(x, y) \to (x + 8, y + 7)$.

    Image: $\underline{\hspace{7cm}}$

76. $\triangle DEF$: $D(1, -6)$, $E(5, -6)$, $F(3, -3)$, translated $8$ units up.

    Rule: $\underline{\hspace{4cm}}$ Image: $\underline{\hspace{6cm}}$

77. Square $(2, 2)$, $(2, 6)$, $(6, 6)$, $(6, 2)$ under $(x, y) \to (x - 9, y - 4)$.

    Image: $\underline{\hspace{7cm}}$

78. Quadrilateral $(-3, 3)$, $(0, 6)$, $(3, 3)$, $(0, 0)$ under $(x, y) \to (x + 4, y - 7)$.

    Image: $\underline{\hspace{7cm}}$

PAGE 20 — Review Part B, continued

Chapter 13 Review · Part B (continued)

FIGURE: fig10-blank-translation-grids.png (full width — one grid per item)

79. $\triangle ABC$: $A(4, -1)$, $B(7, -1)$, $C(7, -4)$ under $(x, y) \to (x - 10, y + 6)$.

    Image: $\underline{\hspace{7cm}}$

80. $\triangle JKL$: $J(-7, 1)$, $K(-4, 1)$, $L(-6, 5)$ under $(x, y) \to (x + 9, y - 6)$.

    Image: $\underline{\hspace{7cm}}$

    $JK = \underline{\hspace{1.5cm}}$ $J'K' = \underline{\hspace{1.5cm}}$ What does the comparison show? $\underline{\hspace{4cm}}$

81. Apply it. A mural design has corners $(1, 1)$, $(5, 1)$, $(3, 4)$ on a grid of one-foot squares, moved $2$ feet right and $3$ feet up.

    Rule: $\underline{\hspace{4cm}}$ New corners: $\underline{\hspace{6cm}}$

82. Find the error. A student plots all four image vertices correctly but connects them in the order $A'$, $C'$, $B'$, $D'$.

    Why is the drawing not the image? _______________________________________________

    Rule for connecting image vertices: _______________________________________________

PAGE 21 — Review Part C

Chapter 13 Review · Part C

Mixed practice and reasoning

FIGURE: fig10-blank-translation-grids.png (full width — grids a and b for items 83 and 88)

83. $\triangle ABC$: $A(-5, -2)$, $B(-1, -2)$, $C(-1, 3)$, translated $4$ units right and $5$ units up. Sketch both figures on grid a).

    Rule: $\underline{\hspace{4cm}}$ Image: $\underline{\hspace{6cm}}$

FIGURE: fig9-review-translation-set.png (half width — grid a for item 84)

84. Grid a). Rule: $\underline{\hspace{4cm}}$ In words: $\underline{\hspace{4cm}}$

    Second pair of vertices that confirms it: $\underline{\hspace{4cm}}$

85. Apply it. A pallet has corners $(6, 2)$, $(9, 2)$, $(9, 4)$, $(6, 4)$. A forklift moves it so the corner at $(6, 2)$ ends at $(1, 7)$.

    Rule: $\underline{\hspace{4cm}}$ Other three corners after the move: $\underline{\hspace{6cm}}$

86. Find the error. A student translates $(-3, 5)$ by "$4$ units left and $2$ units down" and writes $(1, 7)$.

    Sign error in $x$: _______________________________________________

    Sign error in $y$: _______________________________________________

    Correct image: $\underline{\hspace{3cm}}$

87. Explain. Why are the segments joining each preimage vertex to its image all the same length and all pointing the same way?

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    What would a figure look like if that were not true? _______________________________________________

88. Your turn. Choose a triangle with integer vertices and a rule with both a horizontal and a vertical part. Sketch both triangles on grid b).

    My triangle: $\underline{\hspace{6cm}}$ My rule: $\underline{\hspace{4cm}}$

    Image: $\underline{\hspace{6cm}}$

    My congruence check: _______________________________________________