Understand Similarity Through Transformations - 8th Grade Math

1. Identify the missing original side: if $k=\frac{3}{4}$ and the image side is $9$, what was the original side?

   $12$. Divide image by scale factor: $9 ÷ \frac{3}{4} = 12$.

2. What is the effect of a dilation with scale factor $k$ on all lengths?

   All lengths are multiplied by $k$. Dilation multiplies every distance by the scale factor.

3. Identify the scale factor from a side changing from $6$ to $15$.

   $k=\frac{15}{6}=\frac{5}{2}$. Divide new length by original: $\frac{15}{6}=\frac{5}{2}$.

4. Identify a valid sequence to map $A(1,2)$ to $A'(2,4)$ using similarity transformations.

   Dilation about the origin with $k=2$. Each coordinate doubles: $(1×2, 2×2) = (2,4)$.

5. What sequence describes mapping a figure to a similar one: move, turn, flip, then resize?

   Rigid motions (translate/rotate/reflect), then a dilation. First position correctly, then scale to match size.

6. What is always true about corresponding side lengths in similar figures?

   They are proportional by the same scale factor $k$. All sides scale by the same ratio in similar figures.

7. What is the image length after dilation with $k=\frac{3}{2}$ of a segment of length $8$?

   $12$. Multiply original length by scale factor: $8 \times \frac{3}{2} = 12$.

8. What is the scale factor $k$ in a dilation?

   The multiplier applied to all distances from the center. Each point's distance from center is multiplied by $k$.

9. Identify whether triangles with sides $2,3,4$ and $4,6,9$ are similar.

   No, side ratios are not all equal. $\frac{4}{2}=2$, $\frac{6}{3}=2$, but $\frac{9}{4}=2.25$ differs.

10. Which option best describes a valid similarity sequence from figure $A$ to $B$ when $B$ is a rotated and enlarged copy?

    Rotate, then dilate (translation may be included to reposition). Order matters: rotate first to align, then scale up.

11. What is true about corresponding angles in similar figures?

    Corresponding angles are congruent. Dilations preserve angle measures, so angles stay equal.

12. What does a dilation with scale factor $0<k<1$ do to a figure?

    It produces a reduction (smaller similar image). Multiplying by a fraction less than 1 shrinks the figure.

13. Find the image of point $U(2,7)$ after a $90^ irc$ counterclockwise rotation about the origin.

    $U'(-7,2)$. $90°$ CCW rotation maps $(x,y)$ to $(-y,x)$.

14. What is the coordinate rule for a reflection across the $x$-axis?

    $(x,y)\rightarrow(x,-y)$. Reflection across $x$-axis negates the $y$-coordinate.

15. Identify whether the figures are similar if angle measures are $40^\circ,60^\circ,80^\circ$ and $40^\circ,60^\circ,80^\circ$.

    Yes, the angles match (AAA similarity). Same angles guarantee similarity for triangles.

16. Find the image of point $S(-2,4)$ after reflection across the $x$-axis.

    $S'(-2,-4)$. Reflection across $x$-axis negates the $y$-coordinate only.

17. What is the image of $Q(-4,1)$ after a $90^\circ$ counterclockwise rotation about the origin?

    $Q'(-1,-4)$. 90° CCW rotation: $(x,y) \rightarrow (-y,x)$.

18. Find the image of point $R(5,-1)$ after translation by $\langle -3,7 \rangle$.

    $R'(2,6)$. Add translation vector to coordinates: $(5+(-3), -1+7)$.

19. Identify the image length if a segment of length $8$ is dilated by $k=\frac{3}{2}$.

    $12$. Multiply original by scale factor: $8 \times \frac{3}{2} = 12$.

20. What is the scale factor of a dilation that halves every length?

    $k=\frac{1}{2}$. Each distance is multiplied by $\frac{1}{2}$ from the center.

21. What is the image of point $(3,-1)$ after a translation by $(5,4)$?

    $(8,3)$. Add translation vector: $(3+5, -1+4) = (8, 3)$.

22. Find the image of point $T(3,-5)$ after reflection across the $y$-axis.

    $T'(-3,-5)$. Reflection across $y$-axis negates the $x$-coordinate only.

23. What is the effect of a dilation with scale factor $k$ on perimeter?

    Perimeter is multiplied by $k$. Since all sides scale by $k$, perimeter also scales by $k$.

24. What is always true about corresponding angles in similar figures?

    Corresponding angles are congruent. Similarity preserves angle measures between figures.

25. What transformation changes a figure's size but keeps its shape?

    Dilation. Scales all distances by the same factor from a center point.

26. What is true about angle measures in similar figures after transformations and dilation?

    All corresponding angles are equal. Dilations and rigid motions preserve angle measures.

27. What is the coordinate rule for a $90^\circ$ counterclockwise rotation about the origin?

    $(x,y)\rightarrow(-y,x)$. 90° CCW rotation swaps coordinates and negates new $x$.

28. Identify whether rectangles with sides $3,5$ and $6,10$ are similar.

    Yes, because $\frac{6}{3}=\frac{10}{5}=2$. Both ratios equal 2, so sides are proportional.

29. What is the effect of a dilation with scale factor $k$ on area?

    Area is multiplied by $k^2$. Area scales by the square of the linear scale factor.

30. Identify the scale factor from side lengths $18$ (original) and $12$ (image).

    $k=\frac{12}{18}=\frac{2}{3}$. Divide image length by original length: $\frac{12}{18}=\frac{2}{3}$.

31. Find the image of point $P(2,-3)$ after dilation about the origin with $k=4$.

    $P'(8,-12)$. Multiply each coordinate by scale factor: $(2×4, -3×4)$.

32. What is true about corresponding side lengths in similar figures?

    Corresponding sides are proportional by a common factor. Each side ratio equals the scale factor $k$.

33. Which sequence maps $A(0,0)$ to $A'(4,-1)$ using one rigid motion?

    Translate right $4$ and down $1$. Add 4 to x-coordinate and subtract 1 from y-coordinate.

34. What is the image of point $(-2,5)$ after dilation about the origin with $k=3$?

    $(-6,15)$. Apply dilation rule: $(-2×3, 5×3) = (-6, 15)$.

35. What is the image of $P(2,-3)$ after reflection across the $x$-axis?

    $P'(2,3)$. Reflection across x-axis negates the y-coordinate.

36. What is the effect of a reflection on a figure's size and shape?

    Size and shape stay the same; the figure is flipped. Reflection mirrors across a line, preserving all measurements.

37. What is the definition of scale factor in a dilation?

    The ratio $\frac{\text{image length}}{\text{preimage length}}$. Compares new length to original length after dilation.

38. What stays the same under translations, rotations, and reflections?

    Lengths and angle measures. Rigid motions preserve distances and angles.

39. What transformation changes size but preserves shape and angle measures?

    A dilation. Scales all distances by the same factor from a center point.

40. What does it mean for two $2$D figures to be similar using transformations?

    One can be mapped to the other by rigid motions and a dilation. Similar figures have the same shape but may differ in size.

41. What is true about side lengths in similar figures with scale factor $k$?

    Corresponding side lengths have ratio $k$. Dilation multiplies all lengths by the scale factor.

42. What must be true about corresponding side lengths in similar figures?

    Their ratios are equal to the scale factor $k$. All corresponding lengths scale by the same factor in similar figures.

43. What is the scale factor from a figure with side $10$ to a similar figure with corresponding side $4$?

    $k=\frac{4}{10}=\frac{2}{5}$. Scale factor = image length ÷ original length.

44. What transformation changes size but keeps shape and angle measures the same?

    A dilation. Scales all distances from center by same factor, preserving angles.

45. What does it mean for two $2$-D figures to be similar using transformations?

    One can be mapped to the other by rigid motions and a dilation. Rigid motions preserve shape; dilation changes size proportionally.

46. Which transformations are rigid motions (do not change size): rotation, reflection, translation, dilation?

    Rotation, reflection, and translation. These preserve distances and angles; dilation changes size.

47. What is the image of point $ (4,-6) $ after dilation about the origin with $k=\frac{1}{2}$?

    $ (2,-3) $. Apply dilation rule: $(4 × \frac{1}{2}, -6 × \frac{1}{2}) = (2, -3)$

48. What does a dilation with scale factor $k$ do to all lengths in a figure?

    It multiplies every length by $k$. Dilation scales all distances from center by the same factor.

49. What is the image of point $(2,7)$ after a $90^\circ$ counterclockwise rotation about the origin?

    $(-7,2)$. Apply rotation rule: $(2, 7) → (-7, 2)$.

50. Which option describes a valid similarity transformation sequence?

    Rigid motions followed by a dilation (in any order). Combines shape-preserving moves with size change.

51. What is the scale factor of a dilation that doubles every length?

    $k=2$. Each distance is multiplied by 2 from the center.

52. What is the effect of a rotation on a figure's size and shape?

    Size and shape stay the same; orientation changes. Rotation turns around a point without changing distances or angles.

53. What does a dilation with scale factor $k=1$ do to a figure?

    It leaves the figure unchanged (congruent image). Multiplying by 1 preserves all distances.

54. What does a negative dilation scale factor $k<0$ do in addition to resizing?

    It includes a $180^ irc$ turn (a point reflection) about the center. Negative scale factor both resizes and rotates $180°$ around center.

55. What must be true about corresponding angles in similar figures?

    All corresponding angles are congruent. Similarity preserves angle measures between corresponding parts.

56. Identify the area scale factor if similar figures have $k=5$.

    $25$. Area scale factor is $k^2 = 5^2 = 25$.

57. What is the coordinate rule for a translation by $(a,b)$?

    $(x,y)\rightarrow(x+a,y+b)$. Add translation vector to each coordinate.

58. Which transformations are rigid motions in the similarity definition?

    Translations, rotations, and reflections. These preserve distance and angle measures without changing size.

59. Identify a valid sequence to map $B(1,0)$ to $B'(0,1)$ using rigid motions only.

    Rotate $90^ irc$ counterclockwise about the origin. $90°$ CCW rotation transforms $(x,y)$ to $(-y,x)$.

60. Identify the missing side: if $k=2$ and an original side is $7$, what is the image side length?

    $14$. Multiply original length by scale factor: $7 × 2 = 14$.

61. Identify the scale factor $k$ if a side changes from $6$ to $15$ under dilation.

    $k=\frac{15}{6}=\frac{5}{2}$. Scale factor equals new length divided by original length.

62. What is the coordinate rule for a dilation about the origin with scale factor $k$?

    $(x,y)\rightarrow(kx,ky)$. Each coordinate is multiplied by the scale factor.

63. Which option is a correct similarity statement: scale factor $k=1$ or $k=-2$?

    $k=1$. Scale factors must be positive; $k=1$ means congruent figures.

64. Identify whether triangles with sides $3,4,5$ and $6,8,10$ are similar.

    Yes, scale factor $k=2$. All ratios $\frac{6}{3}=\frac{8}{4}=\frac{10}{5}=2$ are equal.

65. What is the scale factor from a figure with side $6$ to a similar figure with corresponding side $9$?

    $k=\frac{9}{6}=\frac{3}{2}$. Scale factor = image length ÷ original length.

66. Identify the scale factor $k$ if a side changes from $12$ to $9$ under dilation.

    $k=\frac{9}{12}=\frac{3}{4}$. Divide image length by pre-image length to find scale factor.

67. Identify the scale factor from side lengths $6$ (original) and $15$ (image).

    $k=\frac{15}{6}=\frac{5}{2}$. Divide image length by original length: $\frac{15}{6}=\frac{5}{2}$.

68. What is the effect of a translation on a figure's size and shape?

    Size and shape stay the same; the figure only shifts. Translation is a rigid motion that slides without rotating or resizing.

69. Identify the perimeter scale factor if figures are similar with $k=\frac{4}{3}$.

    $\frac{4}{3}$. Perimeter scales the same as side lengths.

70. What does it mean for two figures to be similar using transformations?

    One can be mapped to the other by rigid motions and a dilation. Similar figures have same shape but different size via transformations.

71. Identify a valid sequence to map $\triangle ABC$ to a similar $\triangle A'B'C'$ with $k=2$.

    Translate/rotate/reflect to align, then dilate by $k=2$. Position triangles first, then double all lengths.

72. Find the image of point $Q(-6,10)$ after dilation about the origin with $k=\frac{1}{2}$.

    $Q'(-3,5)$. Multiply coordinates by $\frac{1}{2}$: $(-6×\frac{1}{2}, 10×\frac{1}{2})$.

73. Which transformations are rigid motions used in similarity?

    Translations, rotations, and reflections. These preserve shape and size, unlike dilation which changes size.

74. What is the original length if the image length is $14$ under dilation with $k=2$?

    $7$. Divide image length by scale factor: $14 \div 2 = 7$.

75. Which transformations are rigid motions that preserve distances and angles?

    Translations, rotations, and reflections. These preserve congruence without changing size or shape.

76. What does a dilation with scale factor $k>1$ do to a figure?

    It produces an enlargement (larger similar image). Multiplying by a number greater than 1 stretches the figure.

77. What is the coordinate rule for a reflection across the $y$-axis?

    $(x,y)\rightarrow(-x,y)$. Reflection across $y$-axis negates the $x$-coordinate.

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