Question

rigid transformations: preserved properties
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 rigid transformation

Rigid transformation (rotation in this case) preserves the length of the segment. So, the length of \(\overline{AB}\) and \(\overline{A'B'}\) is the same.

Step2: Analyze option A

After a \(90^{\circ}\) rotation, \(\overline{AB}\) (which is parallel to \(x -\)axis) and \(\overline{A'B'}\) will not both be perpendicular to the \(x -\)axis.

Step3: Analyze option B

Since \(\overline{A'B'}\) is the rotation of \(\overline{AB}\) about point \(P\), \(A\) and \(A'\) do not have the same coordinates.

Answer:

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