Question

segment \\(\overline{ab}\\) is the result of rotating \\(\overline{ab}\\) by \\(90^\circ\\) about point \\(p\\).

coordinate plane image with point p at (-4, -1), point b at (-2, -2), point a at (1, -2)

select all of the correct statements about the unchanged properties of \\(\overline{ab}\\) and \\(\overline{ab}\\).

choose all answers that apply:

a \\(\overline{ab}\\) and \\(\overline{ab}\\) are both perpendicular to the \\(x\\)-axis.

b \\(a\\) and \\(a\\) have the same coordinates.

c \\(\overline{ab}\\) and \\(\overline{ab}\\) have the same lengths.

d none of the above

Explanation:

Step1: Analyze Rotation Properties

Rotation is a rigid transformation, which means it preserves length (distance between points) and angle measures, but changes position. So, the length of a segment remains the same after rotation.

Step2: Evaluate Option A

Original segment \( \overline{AB} \) is horizontal (parallel to x - axis), so it's not perpendicular to x - axis. After a \( 90^{\circ} \) rotation, \( \overline{A'B'} \) will be vertical (perpendicular to x - axis). So, \( \overline{AB} \) is not perpendicular to x - axis, so option A is wrong.

Step3: Evaluate Option B

Rotation changes the position of points, so the coordinates of \( A \) and \( A' \) will be different. So, option B is wrong.

Step4: Evaluate Option C

Since rotation is a rigid transformation, the length of \( \overline{AB} \) and \( \overline{A'B'} \) will be the same. So, option C is correct.

Step5: Evaluate Option D

Since option C is correct, option D is wrong.

Answer:

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