Understand Line Segment Transformation Properties - 8th Grade Math

1. What does it mean to say a line segment maps to a segment of the same length under a rigid motion?

   The image segment is congruent to the original. Same length means equal measures: $AB = A'B'$.

2. What always happens to a line after a rotation, reflection, or translation?

   It maps to a line. Rigid motions preserve collinearity of points.

3. Identify the image of $F(-6, 2)$ after reflecting across the $y$-axis.

   $F'(6, 2)$. Apply $(x,y) o (-x,y)$: $(-6,2) o (6,2)$.

4. What is the rule for a $90^\circ$ counterclockwise rotation about the origin for $(x, y)$?

   $(x, y) \to (-y, x)$. Swap coordinates and negate the new $x$.

5. Identify the transformation that slides a figure without turning or flipping it.

   Translation. Moves every point the same distance and direction.

6. Identify the transformation: if points turn around a fixed point by a given angle, what is it?

   A rotation. Points maintain distance from the center of rotation.

7. Identify whether this statement is true or false: A rigid motion can map a line to a segment.

   False. Lines are infinite; segments are finite portions.

8. Find and correct the error: A student claims a translation changes $\overline{AB}$ from length $5$ to length $6$.

   Correct: the image length stays $5$. Translations are rigid motions that preserve length.

9. Identify whether this claim is true: A reflection can change the length of a segment.

   False. Reflections are rigid motions that preserve length.

10. Which option must be true after a rigid motion: $AB = A'B'$ or $AB \ne A'B'$?

    $AB = A'B'$. Equal lengths show distance preservation.

11. What type of transformation is a rigid motion that preserves distances and angle measures?

    A rigid motion (isometry), such as a translation, rotation, or reflection. These transformations preserve all distances and angles.

12. Which rigid motion flips a figure across a line, creating a mirror image?

    Reflection. Each point maps to its mirror image across the line.

13. What happens to the length of a segment under a translation?

    The length stays the same. Translations preserve all distances between points.

14. Find the image length: if $AB = 7$ and a rigid motion maps $A \to A'$ and $B \to B'$, what is $A'B'$?

    $A'B' = 7$. Rigid motions preserve all segment lengths.

15. What happens to the length of a segment under a reflection?

    The length stays the same. Reflections preserve all distances between points.

16. Identify the key property of rigid motions using notation: if $A \to A'$ and $B \to B'$, what is true about $\overline{AB}$ and $\overline{A'B'}$?

    $AB = A'B'$. Rigid motions preserve distances between all point pairs.

17. Identify the image of a segment: if $\overline{AB}$ is transformed, what is its image called?

    The segment $\overline{A'B'}$. Prime notation indicates the transformed version.

18. What happens to the length of a segment under a rotation?

    The length stays the same. Rotations preserve all distances between points.

19. Find the length of $A'B'$ if $AB = 7$ and $AB$ is translated to $A'B'$.

    $7$. Translation preserves all distances.

20. Which transformations are rigid motions: rotation, reflection, translation, dilation?

    Rotation, reflection, and translation. These three preserve distances; dilation changes them.

21. What is the definition of congruent segments using distance notation?

    Segments are congruent if their lengths are equal, like $AB = A'B'$. Congruent means same size and shape.

22. Identify the image of $A(2, -1)$ after translating by $\langle 3, 4 \rangle$.

    $A'(5, 3)$. Add the translation vector: $(2+3, -1+4) = (5, 3)$.

23. What does a line segment map to under any rigid motion (rotation, reflection, or translation)?

    A congruent line segment. Rigid motions preserve both shape and size.

24. Identify the image of $C(4, 1)$ after a $90^\circ$ counterclockwise rotation about the origin.

    $C'(-1, 4)$. Apply $(x,y) o (-y,x)$: $(4,1) o (-1,4)$.

25. What is the relationship between parallel lines after a rigid motion?

    Parallel lines map to parallel lines. Rigid motions preserve angle relationships.

26. Find the length of $A'B'$ if $AB = 4.5$ and $AB$ is rotated to $A'B'$.

    $4.5$. Rotation preserves all distances.

27. What must a rotation, reflection, or translation always map a line segment to?

    A line segment of the same length. Rigid motions preserve segment lengths.

28. Find the length of $A'B'$ if $AB = 12$ and $AB$ is reflected to $A'B'$.

    $12$. Reflection preserves all distances.

29. Identify the transformation: if points flip across a line (mirror line), what is it?

    A reflection. Each point maps to its mirror image across the line.

30. What is the name of a transformation that preserves distances and angle measures?

    A rigid transformation (isometry). Preserves all distances and angles between points.

31. Identify the image of $B(-3, 5)$ after translating by $\langle -2, 1 \rangle$.

    $B'(-5, 6)$. Add the translation vector: $(-3+(-2), 5+1) = (-5, 6)$.

32. Choose the correct statement: If $A$ maps to $A'$ and $B$ maps to $B'$ by a rigid motion, what is true?

    $AB = A'B'$ and $\overline{AB}$ maps to $\overline{A'B'}$. Rigid motions preserve segment lengths and map segments to segments.

33. Identify the image of $E(5, -7)$ after reflecting across the $x$-axis.

    $E'(5, 7)$. Apply $(x,y) o (x,-y)$: $(5,-7) o (5,7)$.

34. Which rigid motion turns a figure around a fixed point by a given angle?

    Rotation. Points rotate by the same angle around a center.

35. What must a rotation, reflection, or translation always preserve about any line segment?

    The segment length stays the same. Rigid motions preserve all distances between points.

36. Identify the image of $D(-2, 3)$ after a $180^\circ$ rotation about the origin.

    $D'(2, -3)$. Apply $(x,y) o (-x,-y)$: $(-2,3) o (2,-3)$.

37. What is the rule for reflecting a point $(x, y)$ across the $x$-axis?

    $(x, y) \to (x, -y)$. Keep $x$ the same, negate $y$.

38. What must a rotation, reflection, or translation always map a line to?

    A line. Lines remain lines under rigid transformations.

39. What is the rule for a $180^\circ$ rotation about the origin for $(x, y)$?

    $(x, y) \to (-x, -y)$. Both coordinates change sign.

40. Which option is true for a line after a rigid motion: it becomes a line or it becomes a curve?

    It becomes a line. Lines map to lines under rigid motions.

41. Find and correct the error: "A translation can map a line to a line segment."

    Correct: A translation maps a line to a line. Lines have infinite length; they stay lines.

42. What is the relationship between slopes of a line and its image after a translation?

    The slopes are equal. Translation doesn't change direction or angle.

43. What is the relationship between perpendicular lines after a rigid motion?

    Perpendicular lines map to perpendicular lines. Right angles remain right angles after rigid motion.

44. Which option is true for a segment after a rigid motion: it keeps length or it changes length?

    It keeps the same length. Rigid motions preserve all distances.

45. What is the correct term for two segments with the same length?

    Congruent segments. Same length means the segments are congruent.

46. What is the rule for reflecting a point $(x, y)$ across the $y$-axis?

    $(x, y) \to (-x, y)$. Keep $y$ the same, negate $x$.

47. Identify the transformation that flips a figure across a line (the line of reflection).

    Reflection. Each point maps to its mirror image across the line.

48. Choose the correct statement: a rigid motion can map a line segment to a ray, yes or no?

    No, it maps to a line segment. Segments stay segments; rays stay rays.

49. What is the relationship between lengths $AB$ and $A'B'$ after a rigid motion?

    $AB = A'B'$. Rigid motions preserve all distances.

50. Identify the transformation: if every point moves the same distance in the same direction, what is it?

    A translation. All points slide the same vector distance.

51. What always happens to a line segment after a rotation, reflection, or translation?

    It maps to a line segment of the same length. Rigid motions preserve distances between endpoints.

52. What does a line map to under any rigid motion (rotation, reflection, or translation)?

    A line. Rigid motions preserve the linear structure of lines.

53. Which option is true for a rigid motion: it preserves length, it changes length, it scales length?

    It preserves length. Definition of rigid motion: distance-preserving.

54. Which option describes the image of a line segment under a rigid motion: ray, segment, circle, parabola?

    A line segment. Rigid motions preserve shape and size.

55. Identify the transformation that turns a figure around a fixed point by a given angle.

    Rotation. Points move along circular arcs around the center.

56. Which transformations are rigid motions in Grade $8$: rotation, reflection, translation, dilation?

    Rotation, reflection, and translation. Dilation changes size, so it's not rigid.

57. Identify whether this statement is true or false: A rigid motion can map a segment to a ray.

    False. Segments map to segments of equal length, not rays.

58. Which rigid motion slides every point the same distance in the same direction?

    Translation. All points move by the same vector.

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