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

coordinate plane image with points p, b, a and segment ab, then question about unchanged properties of ab and ab with options a to d

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

Question

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

coordinate plane image with points p, b, a and segment ab, then question about unchanged properties of ab and ab with options a to d

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

Accepted Answer

# Brief Explanations:
- **Option A**: $\overline{AB}$ is horizontal (parallel to the x - axis), not perpendicular. After a $90^{\circ}$ rotation, $\overline{A'B'}$ will be vertical (perpendicular to the x - axis). So this statement is false.
- **Option B**: A rotation about point $P$ will change the coordinates of $A$ to get $A'$, so $A$ and $A'$ do not have the same coordinates. This statement is false.
- **Option C**: Rotations are rigid transformations, which means the length of a segment remains unchanged after rotation. So $\overline{AB}$ and $\overline{A'B'}$ have the same length. This statement is true.
- **Option D**: Since option C is correct, this option is false.

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

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