Congruence via Rigid Motions - Geometry

1. What is an invariant under any rigid motion?

   Distance between any two points. Rigid motions preserve all distances by definition.

2. What is the image of $\left(-6,2\right)$ after a $180^\circ$ rotation about the origin?

   $\left(6,-2\right)$. Both coordinates change sign for $180°$ rotation.

3. Identify whether the mapping preserves orientation: $\left(x,y\right)\to\left(x+2,y-3\right)$.

   Preserves orientation. Translation maintains clockwise/counterclockwise order.

4. Identify whether the mapping preserves orientation: $\left(x,y\right)\to\left(x,-y\right)$.

   Reverses orientation. Reflection flips orientation from clockwise to counterclockwise.

5. What is the image of $\left(4,-1\right)$ after a $90^\circ$ clockwise rotation about the origin?

   $\left(-1,-4\right)$. Apply the $90°$ clockwise rotation rule.

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

   $\left(x,y\right)\to\left(y,-x\right)$. This is the $90°$ clockwise rotation transformation rule.

7. Which rigid motion is determined by a center point and an angle of rotation?

   A rotation. Rotations require a center point and angle measure.

8. What is the coordinate rule for reflection across the line $y=x$?

   $\left(x,y\right)\to\left(y,x\right)$. The coordinates swap positions across the diagonal line $y=x$.

9. What is the coordinate rule for reflection across the line $y=-x$?

   $\left(x,y\right)\to\left(-y,-x\right)$. Both coordinates change sign and swap positions across $y=-x$.

10. Which rigid motion maps every point on a line to a point on a parallel line?

    A translation. Translation moves all points the same distance in the same direction.

11. What is the image of $\left(3,-4\right)$ after reflection across the $x$-axis?

    $\left(3,4\right)$. Reflection across $x$-axis changes $y$-coordinate sign.

12. Which reflection maps $\left(3,5\right)$ to $\left(3,-5\right)$?

    Reflection across the $x$-axis. The $x$-coordinates are equal, so reflect across $x$-axis.

13. Which rigid motion is determined by a line of reflection?

    A reflection. Reflections require a line to reflect across.

14. What is the definition of the preimage in a transformation problem?

    The original figure before the transformation. The preimage is the starting figure before transformation.

15. What happens to the area of a figure under a rigid motion?

    Area is unchanged. Rigid motions preserve shape and size, so area stays constant.

16. What is the definition of a rigid motion in the plane?

    A transformation that preserves all distances and angle measures. This is the fundamental property that defines rigid motions.

17. What is the image of $\left(1,7\right)$ after reflection across the line $y=x$?

    $\left(7,1\right)$. Coordinates swap when reflecting across $y=x$.

18. What is the image of $\left(9,-2\right)$ after reflection across the line $y=1$?

    $\left(9,4\right)$. Use the reflection formula $(x,y) \to (x,2b-y)$ with $b=1$.

19. What happens to the perimeter of a figure under a rigid motion?

    Perimeter is unchanged. Rigid motions preserve all lengths, so perimeter stays constant.

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

    One can be mapped onto the other by a sequence of rigid motions. Congruence is defined as the ability to map one figure onto another using rigid motions.

21. What is the image of $\left(-2,5\right)$ after reflection across the $y$-axis?

    $\left(2,5\right)$. Reflection across $y$-axis changes $x$-coordinate sign.

22. Identify the rigid motion: $\left(x,y\right)\to\left(-x,-y\right)$.

    $180^\circ$ rotation about the origin. Both coordinates change sign for $180°$ rotation.

23. Which rigid motion reverses orientation (clockwise becomes counterclockwise)?

    A reflection. Reflection flips the figure, reversing vertex order.

24. Which rigid motion always preserves orientation (clockwise order of vertices)?

    A translation or a rotation. These motions maintain the order of vertices around a figure.

25. Identify the rigid motion: $\left(x,y\right)\to\left(-y,x\right)$.

    $90^\circ$ counterclockwise rotation about the origin. This transformation rotates points $90°$ counterclockwise.

26. What is the definition of the image in a transformation problem?

    The resulting figure after the transformation. The image is the final figure after transformation.

27. What is the coordinate rule for a translation right $h$ and up $k$?

    $\left(x,y\right)\to\left(x+h,y+k\right)$. Standard translation rule moving right $h$ units and up $k$ units.

28. What is the coordinate rule for reflection across the vertical line $x=a$?

    $\left(x,y\right)\to\left(2a-x,y\right)$. Reflects across a vertical line by using the distance formula.

29. What is the coordinate rule for a $180^\circ$ rotation about the origin?

    $\left(x,y\right)\to\left(-x,-y\right)$. Half rotation flips both coordinates through the origin.

30. What three transformations are the basic rigid motions in the plane?

    Translations, rotations, and reflections. These are the only transformations that preserve distance and angle measure.

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

    $\left(x,y\right)\to\left(-y,x\right)$. Standard rotation matrix for $90°$ counterclockwise about the origin.

32. Which rigid motion is determined by a translation vector $\langle a,b\rangle$?

    A translation. Translations require a direction and distance vector.

33. What is the coordinate rule for a translation by vector $\langle a,b\rangle$?

    $\left(x,y\right)\to\left(x+a,y+b\right)$. Add the components of the translation vector to the coordinates.

34. What is the result of reflecting across the $x$-axis, then across the $y$-axis?

    A $180^\circ$ rotation about the origin. Composition of reflections across perpendicular axes.

35. What is the definition of the image of a point $P$ under a transformation?

    The point $P'$ obtained after applying the transformation to $P$. The image is the result after applying the transformation.

36. Identify the rigid motion: $\left(x,y\right)\to\left(x,-y\right)$.

    Reflection across the $x$-axis. The $y$-coordinate changes sign while $x$ remains unchanged.

37. Which rotation about the origin maps $\left(2,-7\right)$ to $\left(-2,7\right)$?

    $180^\circ$ rotation about the origin. Both coordinates change sign for $180°$ rotation.

38. What is the image of $\left(2,3\right)$ after translation by $\langle -4,6\rangle$?

    $\left(-2,9\right)$. Add the translation vector components to the coordinates.

39. Which reflection maps $\left(3,5\right)$ to $\left(-3,5\right)$?

    Reflection across the $y$-axis. The $y$-coordinates are equal, so reflect across $y$-axis.

40. Identify the rigid motion: $\left(x,y\right)\to\left(y,x\right)$.

    Reflection across the line $y=x$. Coordinates swap positions in this reflection transformation.

41. Which statement is always true if figures $F$ and $G$ are congruent by rigid motions?

    Corresponding side lengths and angle measures are equal. Congruent figures have identical measurements.

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

    $\left(1,4\right)$. Apply the $90°$ counterclockwise rotation rule.

43. Identify the rigid motion: $\left(x,y\right)\to\left(-x,y\right)$.

    Reflection across the $y$-axis. The $x$-coordinate changes sign while $y$ remains unchanged.

44. Identify the rigid motion: $\left(x,y\right)\to\left(x+5,y-2\right)$.

    Translation by $\langle 5,-2\rangle$. Adding constants to coordinates indicates translation.

45. What is the coordinate rule for reflection across the horizontal line $y=b$?

    $\left(x,y\right)\to\left(x,2b-y\right)$. Reflects across a horizontal line by using the distance formula.

46. Which rigid motion maps $A\left(1,2\right)$ to $A'\left(6,-1\right)$?

    Translation by $\langle 5,-3\rangle$. Find the translation vector by subtracting coordinates.

47. Which rotation about the origin maps $\left(5,1\right)$ to $\left(1,-5\right)$?

    $90^\circ$ clockwise rotation about the origin. Apply the $90°$ clockwise rotation rule.

48. What angle relationship is preserved under any rigid motion?

    Angle measure is preserved. All angle measures remain the same under rigid motions.

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

    $\left(x,y\right)\to\left(x,-y\right)$. The $x$-coordinate stays the same, the $y$-coordinate changes sign.

50. Which rotation about the origin maps $\left(5,1\right)$ to $\left(-1,5\right)$?

    $90^\circ$ counterclockwise rotation about the origin. Apply the $90°$ counterclockwise rotation rule.

51. What is the image of $\left(-1,8\right)$ after reflection across the line $x=2$?

    $\left(5,8\right)$. Use the reflection formula $(x,y) \to (2a-x,y)$ with $a=2$.

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

    $\left(x,y\right)\to\left(-x,y\right)$. The $y$-coordinate stays the same, the $x$-coordinate changes sign.

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