Performing and Sequencing Rigid Transformations - Geometry

1. What is the composition of reflections across $y=x$ then across the $x$-axis?

   Rotation $90^\circ$ clockwise about the origin. Composition of these reflections produces clockwise rotation.

2. What is the image of $(6,3)$ under a $180^\circ$ rotation about $(2,-1)$?

   $(-2,-5)$. Apply rule $(2h-x,2k-y)$ with center $(2,-1)$.

3. What is the composition of reflections across $x$-axis then across $y=x$?

   Rotation $90^\circ$ counterclockwise about the origin. Composition of these reflections produces counterclockwise rotation.

4. What is the image of $(2,-3)$ after $90^\circ$ counterclockwise rotation about the origin?

   $(3,2)$. Apply CCW rotation rule $(-y,x)$: $(-(-3),2) = (3,2)$.

5. What is the coordinate rule for a $180^\circ$ rotation about the point $(h,k)$?

   $(x,y)\to(2h-x,2k-y)$. Rotation about center point by half turn.

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

   $(x,y)\to(-y,x)$. Standard rotation matrix for $90°$ counterclockwise.

7. What is the image of $(4,1)$ under a $90^\circ$ clockwise rotation about the origin?

   $(1,-4)$. Apply rule $(y,-x)$: $(4,1) \to (1,-4)$.

8. Identify the transformation that maps $(x,y)$ to $(-x,-y)$.

   Rotation $180^\circ$ about the origin. Rule $(-x,-y)$ represents $180°$ rotation.

9. What is the image of $(-1,6)$ under reflection across the $x$-axis?

   $(-1,-6)$. Negate $y$-coordinate: $(-1,6) \to (-1,-6)$.

10. What is the coordinate rule for translating $(x,y)$ up $d$ units?

    $(x,y)\to(x,y+d)$. Add positive value to $y$-coordinate for upward movement.

11. What is the image of $(3,-2)$ under translation by $\langle -5,4\rangle$?

    $(-2,2)$. Apply translation rule: $(3+(-5), -2+4) = (-2,2)$.

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

    $(x,y)\to(-y,-x)$. Negate both then swap to reflect across opposite diagonal.

13. Identify the transformation that maps $(x,y)$ to $(x+7,y-3)$.

    Translation by $\langle 7,-3\rangle$. Translation vector is $\langle 7,-3\rangle$ from coordinate rule.

14. What is the image of $(-3,5)$ under reflection across the line $y=2$?

    $(-3,-1)$. Apply rule $(x,2k-y)$ with $k=2$: $(-3,4-5) = (-3,-1)$.

15. Identify the transformation that maps $(x,y)$ to $(y,x)$.

    Reflection across $y=x$. Swapping coordinates indicates reflection across $y=x$.

16. Which rigid motion reverses orientation (clockwise order of vertices)?

    Reflection. Reflections flip orientation while preserving shape.

17. What is the coordinate rule for a $90^\circ$ clockwise rotation about $(h,k)$?

    $(x,y)\to(h+(y-k),k-(x-h))$. Translate to origin, rotate clockwise, translate back.

18. What sequence maps $(x,y)$ to $(2-y,3+x)$ using a $90^\circ$ rotation and a translation?

    Rotate $90^\circ$ CCW about origin, then translate by $\langle 2,3\rangle$. First rotate, then translate by vector components.

19. What is the coordinate rule for reflecting across the vertical line $x=h$?

    $(x,y)\to(2h-x,y)$. Distance from line equals distance on other side.

20. Which rigid motions preserve orientation?

    Translations and rotations. These transformations maintain clockwise vertex ordering.

21. What is the coordinate rule for a rotation about the origin by $360^\circ$?

    $(x,y)\to(x,y)$. Full rotation returns point unchanged.

22. What is the inverse of a reflection across a line $\ell$?

    The same reflection across $\ell$. Reflections are their own inverse transformation.

23. What sequence maps $(x,y)$ to $(5-x,y)$ using a reflection and a translation?

    Reflect across the $y$-axis, then translate by $\langle 5,0\rangle$. First reflect, then translate rightward by 5 units.

24. What is the image of triangle vertex $P(-2,3)$ after reflection across $x=0$?

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

25. What is the image of $(1,4)$ under a $90^\circ$ counterclockwise rotation about $(2,1)$?

    $(-1,0)$. Apply $90°$ CCW rotation formula about $(2,1)$.

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

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

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

    $(x,y)\to(-x,-y)$. Negate both coordinates for half rotation.

28. What is the image of $(2,7)$ under reflection across the line $x=1$?

    $(0,7)$. Apply rule $(2h-x,y)$ with $h=1$: $(2-2,7) = (0,7)$.

29. What is the image of $(2,-3)$ after $90^\circ$ clockwise rotation about the origin?

    $(-3,-2)$. Apply CW rotation rule $(y,-x)$: $(-3,-2)$.

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

    $(x,y)\to(x,-y)$. Negate the $y$-coordinate to flip across horizontal axis.

31. Identify the transformation that maps $(x,y)$ to $(x,-y)$.

    Reflection across the $x$-axis. Negating $y$-coordinate indicates reflection across $x$-axis.

32. Identify the transformation that maps $(x,y)$ to $(-y,x)$.

    Rotation $90^\circ$ counterclockwise about the origin. Rule $(-y,x)$ represents $90°$ counterclockwise rotation.

33. What is the composition of reflections across the $x$-axis then the $y$-axis?

    Rotation $180^\circ$ about the origin. Two reflections across perpendicular lines equals $180°$ rotation.

34. What is the coordinate rule for translating $(x,y)$ left $d$ units?

    $(x,y)\to(x-d,y)$. Subtract positive value from $x$-coordinate for leftward movement.

35. What is the image of $(5,-2)$ under reflection across the line $y=x$?

    $(-2,5)$. Swap coordinates: $(5,-2) \to (-2,5)$.

36. What is the image of $(4,1)$ under a $90^\circ$ counterclockwise rotation about the origin?

    $(-1,4)$. Apply rule $(-y,x)$: $(4,1) \to (-1,4)$.

37. What is the composition of two translations by $\langle a,b\rangle$ and $\langle c,d\rangle$?

    Translation by $\langle a+c,b+d\rangle$. Add corresponding components of translation vectors.

38. What is the coordinate rule for reflecting across the horizontal line $y=k$?

    $(x,y)\to(x,2k-y)$. Distance from line equals distance on other side.

39. Identify the transformation that maps $(x,y)$ to $(-x,y)$.

    Reflection across the $y$-axis. Negating $x$-coordinate indicates reflection across $y$-axis.

40. Identify the transformation that maps $(x,y)$ to $(y,-x)$.

    Rotation $90^\circ$ clockwise about the origin. Rule $(y,-x)$ represents $90°$ clockwise rotation.

41. What is the image of $(5,-2)$ under reflection across the line $y=-x$?

    $(2,-5)$. Negate both then swap: $(5,-2) \to (2,-5)$.

42. Identify the transformation that maps $(x,y)$ to $(-y,-x)$.

    Reflection across $y=-x$. Negating both then swapping indicates reflection across $y=-x$.

43. What is the image of $(4,1)$ under a $180^\circ$ rotation about the origin?

    $(-4,-1)$. Apply rule $(-x,-y)$: $(4,1) \to (-4,-1)$.

44. What is the coordinate rule for a $90^\circ$ counterclockwise rotation about $(h,k)$?

    $(x,y)\to(h-(y-k),k+(x-h))$. Translate to origin, rotate, translate back.

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

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

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

    $(x,y)\to(-x,y)$. Negate the $x$-coordinate to flip across vertical axis.

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

    $(x,y)\to(y,-x)$. Standard rotation matrix for $90°$ clockwise.

48. What is the image of $(1,4)$ under a $90^\circ$ clockwise rotation about $(2,1)$?

    $(5,2)$. Apply $90°$ CW rotation formula about $(2,1)$.

49. What is the image of point $(0,-5)$ after reflection across $y=2$?

    $(0,9)$. Apply reflection rule $(x,2k-y)$ with $k=2$.

50. What is the image of $(-1,6)$ under reflection across the $y$-axis?

    $(1,6)$. Negate $x$-coordinate: $(-1,6) \to (1,6)$.

51. What is the inverse of a $90^\circ$ counterclockwise rotation about the origin?

    A $90^\circ$ clockwise rotation about the origin. Inverse rotation is opposite direction by same angle.

52. What is the image of segment endpoints $A(1,2)$ and $B(4,-1)$ after $\langle 3,5\rangle$?

    $A'(4,7),\ B'(7,4)$. Add translation vector to each endpoint.

53. What is the coordinate rule for translating $(x,y)$ right $d$ units?

    $(x,y)\to(x+d,y)$. Add positive value to $x$-coordinate for rightward movement.

54. Which transformation always preserves distances and angle measures?

    Any rigid motion (translation, reflection, or rotation). Rigid motions preserve all distance and angle measurements.

55. What is the inverse transformation of translation by $\langle a,b\rangle$?

    Translation by $\langle -a,-b\rangle$. Inverse translation uses opposite direction vector.

56. What is the coordinate rule for translating $(x,y)$ down $d$ units?

    $(x,y)\to(x,y-d)$. Subtract positive value from $y$-coordinate for downward movement.

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