Defining Rotations, Reflections, and Translations - Geometry

1. What is the defining distance property of a translation for any points $P$ and $Q$?

   $PQ=P'Q'$ (translations preserve distances). All distances between points remain unchanged.

2. Identify the image of $(2,5)$ under reflection across the line $y=x$.

   $(5,2)$. Line $y=x$ reflection swaps coordinates.

3. What does it mean to say a reflection preserves distance?

   For any $P,Q$, $PQ=P'Q'$ after reflection. All distances between points are preserved.

4. What is a translation in the plane, defined using directed line segments?

   A mapping that moves every point by the same directed segment (vector). Each point moves by the same displacement vector.

5. What geometric locus describes all points $P'$ that a rotation can send a given point $P$ to?

   A circle centered at $O$ with radius $OP$. All points equidistant from center trace circular paths.

6. Identify the image of $(2,5)$ under reflection across the line $y=-x$.

   $(-5,-2)$. Line $y=-x$ reflection swaps and negates coordinates.

7. What must be true about $PP'$ if $P$ is reflected across line $l$ to $P'$?

   Segment $PP'$ is perpendicular to $l$. Reflection requires perpendicularity to mirror line.

8. What is the fixed set of a translation by a nonzero vector?

   No points are fixed (if the vector is nonzero). Every point moves by the same nonzero displacement.

9. What points are fixed (unchanged) by a reflection across line $l$?

   Exactly the points on $l$. Points on the mirror line map to themselves.

10. Identify the rigid motion: a line is the perpendicular bisector of each point-image segment.

    Reflection. The mirror line bisects each point-image segment perpendicularly.

11. Which rigid motion is defined using perpendicular lines to a given line $l$?

    A reflection across $l$. Each point reflects across line $l$ using perpendicular projection.

12. What point is fixed by any rotation about center $O$?

    The center $O$ is fixed. The center of rotation doesn't move during rotation.

13. What is the center of rotation if $P$ maps to $P'$ and the center lies on the perpendicular bisector of $PP'$?

    The center is the intersection of perpendicular bisectors of two pairs $PP'$. Center equidistant from corresponding points in rotation.

14. Identify the image of $(-1,6)$ under reflection across the horizontal line $y=4$.

    $(-1,2)$. Reflect across $y=4$: $y$-coordinate becomes $2(4)-6=2$.

15. Which rigid motion is defined using circles centered at a fixed point $O$?

    A rotation about $O$. Points move along concentric circles around fixed center.

16. Identify the image of $(7,3)$ under reflection across the vertical line $x=2$.

    $(-3,3)$. Reflect across $x=2$: $x$-coordinate becomes $2(2)-7=-3$.

17. Identify the image of $(2,5)$ under reflection across the $y$-axis.

    $(-2,5)$. $y$-axis reflection negates the $x$-coordinate.

18. What does it mean to say a rotation preserves distance?

    For any $P,Q$, $PQ=P'Q'$ after rotation. Rotation is an isometry preserving all distances.

19. What is the sign convention for a positive rotation angle in the coordinate plane?

    Positive angles rotate counterclockwise. Standard mathematical convention for angle measurement.

20. What line must contain the midpoint of $PP'$ for a reflection that maps $P$ to $P'$?

    The mirror line $l$ must pass through the midpoint of $PP'$. Mirror line is perpendicular bisector of segment $PP'$.

21. What must be true about distances to the mirror line in a reflection across $l$?

    $d(P,l)=d(P',l)$. Point and image are equidistant from mirror line.

22. Identify the image of $(2,5)$ under reflection across the $x$-axis.

    $(2,-5)$. $x$-axis reflection negates the $y$-coordinate.

23. What is the definition of the image of point $P$ under reflection across line $l$?

    $P'$ such that $l$ is the perpendicular bisector of segment $PP'$. The mirror line is the perpendicular bisector of segment $PP'$.

24. Which rigid motion is defined using parallel lines through each point?

    A translation (motion along parallel directed segments). All points move along parallel directed line segments.

25. Identify the rigid motion: points move along circles centered at a fixed point.

    Rotation. Points trace circular arcs around the center.

26. Identify the rigid motion: every point moves the same distance in the same direction.

    Translation. Uniform motion without rotation or reflection.

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