Understand Congruence Through Transformations - 8th Grade Math

1. Identify the rigid motion: points keep orientation and slide with no turning or flipping.

   Translation. Sliding preserves orientation without rotation.

2. Identify the transformation: $(x,y)\rightarrow(-y,x)$.

   Rotation $90^\circ$ counterclockwise about the origin. $(x,y) o (-y,x)$ rotates $90°$ counterclockwise.

3. What property of distance is preserved by any rigid motion?

   All segment lengths (distances) stay the same. Rigid motions preserve the shape and size of figures.

4. What happens to orientation (clockwise order of vertices) under a translation or rotation?

   Orientation is preserved. Translations and rotations don't flip vertex order.

5. Find the image of $R(3,-2)$ after a $90^\circ$ counterclockwise rotation about the origin.

   $R'(2,3)$. Apply rule $(x,y)\rightarrow(-y,x)$ to get $(-(-2),3)=(2,3)$.

6. Identify the rigid motion: a figure is flipped so orientation reverses across a line.

   Reflection. Flipping reverses the figure's orientation.

7. Identify the rule for reflecting a point $(x,y)$ across the $x$-axis.

   $(x,y)\rightarrow (x,-y)$. Negate the $y$-coordinate to flip over $x$-axis.

8. What does it mean for two plane figures to be congruent using rigid motions?

   One can be obtained from the other by translations, rotations, and reflections. Rigid motions preserve size and shape while moving figures.

9. Identify the translation that maps $A(-1,2)$ to $A'(3,7)$.

   Translate by $\langle 4,5\rangle$. $3-(-1)=4$ and $7-2=5$ give the translation vector.

10. What is a reflection described in words?

    A flip across a line (the line of reflection). Each point maps to its mirror image.

11. Identify the transformation: $(x,y)\rightarrow(x,-y)$.

    Reflection across the $x$-axis. Negating $y$ flips points across the $x$-axis.

12. Which transformations are rigid motions that preserve distances and angle measures?

    Translation, rotation, and reflection. These transformations preserve distances and angles.

13. Identify the transformation: $(x,y)\rightarrow(x+3,y-2)$.

    Translation right $3$ and down $2$. Add $3$ to $x$, subtract $2$ from $y$.

14. What does a translation do to every point of a figure on the coordinate plane?

    Moves every point the same distance in the same direction. Translation slides without rotating or flipping.

15. What is the minimum information needed to describe a rotation precisely?

    Center of rotation, angle measure, and direction. Must specify where, how much, and which way to rotate.

16. Identify the rule for rotating a point $(x,y)$ by $180^{\circ}$ about the origin.

    $(x,y)\rightarrow (-x,-y)$. Negate both coordinates for half-turn.

17. Find the image of $P(2,-5)$ after reflection across the $x$-axis.

    $P'(2,5)$. Reflecting across $x$-axis changes sign of $y$-coordinate.

18. What is the coordinate rule for reflecting a point across the $x$-axis?

    $(x,y)\rightarrow(x,-y)$. Negate the $y$-coordinate while keeping $x$ unchanged.

19. What is the coordinate rule for a $270^\circ$ counterclockwise rotation about the origin?

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

20. What is the coordinate rule for translating a point by $\langle a,b\rangle$?

    $(x,y)\rightarrow(x+a,y+b)$. Add $a$ to $x$-coordinate and $b$ to $y$-coordinate.

21. What property of angle measure is preserved by any rigid motion?

    All angle measures stay the same. Rigid motions don't change the shape of figures.

22. What is a translation described in words?

    A slide that moves every point the same distance and direction. Points move parallel without rotation.

23. Identify the rule for reflecting a point $(x,y)$ across the $y$-axis.

    $(x,y)\rightarrow (-x,y)$. Negate the $x$-coordinate to flip over $y$-axis.

24. What is the image of a point $P$ after a transformation usually called?

    $P'$ (read as "P prime"). Prime notation indicates the transformed point.

25. What happens to orientation (clockwise order of vertices) under a reflection?

    Orientation is reversed. Reflections flip the order of vertices.

26. Identify the translation rule for moving a point $(x,y)$ right $a$ and up $b$.

    $(x,y)\rightarrow (x+a,y+b)$. Add $a$ to $x$-coordinate, $b$ to $y$-coordinate.

27. Identify the transformation: $(x,y)\rightarrow(y,x)$.

    Reflection across the line $y=x$. Swapping coordinates reflects across $y=x$.

28. What is a rotation described in words?

    A turn around a fixed point (the center of rotation). All points rotate by the same angle.

29. Identify the rule for reflecting a point $(x,y)$ across the line $y=x$.

    $(x,y)\rightarrow (y,x)$. Swap coordinates to reflect across diagonal.

30. What is the image of $P(-2,5)$ after the translation $(x,y)\rightarrow(x+4,y-3)$?

    $P'(2,2)$. $(-2+4, 5-3) = (2,2)$.

31. Which sequence maps $(x,y)$ to $(-x,y)$: reflect across $x$-axis or across $y$-axis?

    Reflect across the $y$-axis. Changing $(x,y)$ to $(-x,y)$ negates only $x$-coordinate.

32. What is a rotation of a figure in the plane?

    A turn about a fixed center by a specified angle and direction. Points rotate around a center point.

33. What is a reflection of a figure in the plane?

    A flip across a line so each point maps to an equal-distance point on the other side. Points mirror across the line of reflection.

34. What is a translation of a figure in the plane?

    A slide that moves every point the same distance in the same direction. Every point moves by the same vector.

35. Which sequence maps $A(1,2)$ to $A'(4,-1)$: translate or rotate about origin?

    Translate by $(+3,-3)$. $(1,2) o (4,-1)$ requires adding $(3,-3)$.

36. What does a reflection do to a figure relative to a line?

    Flips the figure across a line, creating a mirror image. Each point maps to its perpendicular distance on opposite side.

37. What two measurements are preserved by any rigid motion?

    Side lengths and angle measures are preserved. Rigid motions don't stretch or shrink figures.

38. What is a sequence of transformations (a composition) in geometry?

    Two or more transformations applied one after another. The result of the first feeds into the next.

39. Identify the rule for rotating a point $(x,y)$ by $90^{\circ}$ clockwise about the origin.

    $(x,y)\rightarrow (y,-x)$. Swap and negate second coordinate for 90° CW turn.

40. Which transformations are rigid motions that always preserve congruence?

    Translations, rotations, and reflections. These three transformations preserve distances and angles.

41. Identify the transformation: $(x,y)\rightarrow(y,-x)$.

    Rotation $90^\circ$ clockwise about the origin. $(x,y) o (y,-x)$ rotates $90°$ clockwise.

42. Identify the rule for rotating a point $(x,y)$ by $90^{\circ}$ counterclockwise about the origin.

    $(x,y)\rightarrow (-y,x)$. Swap and negate first coordinate for 90° CCW turn.

43. What does a rotation do to a figure relative to a fixed point?

    Turns the figure around a center by a given angle and direction. All points maintain their distance from the center.

44. Identify the transformation: $(x,y)\rightarrow(-x,y)$.

    Reflection across the $y$-axis. Negating $x$ flips points across the $y$-axis.

45. Identify the transformation: $(x,y)\rightarrow(-x,-y)$.

    Rotation $180^\circ$ about the origin. Negating both coordinates rotates $180°$.

46. Which sequence maps a figure to its mirror image across the $y$-axis without changing size?

    Reflect the figure across the $y$-axis. Reflection preserves size while creating mirror image.

47. Find the image of $S(-6,4)$ after a $180^\circ$ rotation about the origin.

    $S'(6,-4)$. Apply rule $(x,y)\rightarrow(-x,-y)$ to get $(-(-6),-4)=(6,-4)$.

48. What is the image of $P(1,6)$ after a $90^\circ$ counterclockwise rotation about the origin?

    $P'(-6,1)$. $90°$ counterclockwise: $(1,6) o (-6,1)$.

49. What property of a figure is preserved by any rigid motion (translation, rotation, reflection)?

    Lengths and angle measures (size and shape) are preserved. Rigid motions don't change measurements.

50. What is the coordinate rule for reflecting a point across the $y$-axis?

    $(x,y)\rightarrow(-x,y)$. Negate the $x$-coordinate while keeping $y$ unchanged.

51. Find the image of $Q(-4,1)$ after reflection across the $y$-axis.

    $Q'(4,1)$. Reflecting across $y$-axis changes sign of $x$-coordinate.

52. What is the coordinate rule for reflecting a point across the line $y=x$?

    $(x,y)\rightarrow(y,x)$. Swap coordinates to reflect across the diagonal.

53. Which statement is always true if figure $A$ is congruent to figure $B$?

    Corresponding sides are equal and corresponding angles are equal. Congruent figures have identical measurements.

54. What is the image of $P(3,-4)$ after reflection across the $y$-axis?

    $P'(-3,-4)$. Reflect across $y$-axis: negate $x$-coordinate.

55. What is the key difference between congruence and similarity?

    Congruence uses rigid motions only; similarity may include dilation. Dilation changes size but rigid motions don't.

56. Which transformations are rigid motions that preserve distance and angle measure?

    Translations, rotations, and reflections. These transformations preserve distances and angles.

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