Question

complete the paragraph to describe a rotation ( x^circ ) counterclockwise around point ( p ). the image of ( p ) is itself. for any other point ( a ), the image of ( a ) is ( a ) where (overline{pa} parallel overline{pa}), (overline{pa} perp overline{pa}), or ( pa = pa ) (dropdown options), and ( mangle apa = ) dropdown.

Explanation:

1. For the first blank (relationship between \( \overline{PA} \) and \( \overline{PA'} \)): In a rotation around point \( P \), the distance from the center of rotation \( P \) to a point \( A \) and its image \( A' \) remains the same. So \( PA = PA' \) (rotation preserves distance from the center). The other options: \( \overline{PA} \parallel \overline{PA'} \) is not true for rotation (unless it's a 0° or 360° rotation, but generally, rotation changes the direction, so they aren't parallel), and \( \overline{PA} \perp \overline{PA'} \) is only true for 90° rotations, not in general for any \( x^\circ \) rotation.
2. For the second blank (measure of \( \angle APA' \)): The angle of rotation is the angle between the original segment \( PA \) and the rotated segment \( PA' \), so \( m\angle APA' = x^\circ \).

Answer:

First blank: \( PA = PA' \)
Second blank: \( x^\circ \)