Question

segment \\( \overline { a ^ { \prime } b ^ { \prime } } \\) is the result of rotating \\( \overline { a b } \\) by \\( 90 ^ { \circ } \\) about point \\( p \\).
select all of the correct statements about the unchanged properties of \\( \overline { a b } \\) and \\( \overline { a ^ { \prime } b ^ { \prime } } \\).
choose all answers that apply:
\\( \square \\) \\( \overline { a b } \\) and \\( \overline { a ^ { \prime } b ^ { \prime } } \\) are both perpendicular to the \\( x \\)-axis.
\\( \square \\) \\( a \\) and \\( a ^ { \prime } \\) have the same coordinates.
\\( \square \\) \\( \overline { a b } \\) and \\( \overline { a ^ { \prime } b ^ { \prime } } \\) have the same lengths.
\\( \square \\) none of the above

Explanation:

Step1: Analyze property of rotation

Rotation is a rigid transformation. Rigid transformations preserve the length of segments.

Step2: Check each option

• Option A: $\overline{AB}$ is parallel to the $x -$axis (horizontal). After a $90^{\circ}$ rotation about point $P$, $\overline{A'B'}$ will be perpendicular to $\overline{AB}$, so $\overline{A'B'}$ is not parallel to the $x -$axis.
• Option B: Rotation about point $P$ (not about point $A$) will change the position of point $A$ (unless $A = P$, which is not the case here), so $A$ and $A'$ do not have the same coordinates.
• Option C: Since rotation is a rigid transformation, the length of $\overline{AB}$ and $\overline{A'B'}$ is preserved.

Answer:

C. $\overline{AB}$ and $\overline{A'B'}$ have the same lengths.