Question

triangle $abc$ is rotated about point $p$. select the true statements.
draw
show your work here
the distance between $p$ and $b$ equals the difference between $p$ and $b$.
$m\angle a = m\angle b$
$m\angle a = m\angle a$
the distance between $p$ and $a$ equals the difference between $p$ and $b$.
$m\angle b = m\angle c$
none of the above

Explanation:

Step1: Recall rotation properties

Rotations preserve side lengths, distances from the center of rotation to corresponding points, and angle measures. Corresponding angles (original and rotated) are congruent, and the distance from the center $P$ to a point equals the distance from $P$ to its rotated image.

Step2: Evaluate each statement

1. "The distance between $P$ and $B$ equals the difference between $P$ and $B'$": False. Rotation preserves distance, so $PB = PB'$, not equal to their difference.
2. $m\angle A = m\angle B'$: False. $\angle A$ corresponds to $\angle A'$, not $\angle B'$, so their measures are not equal.
3. $m\angle A = m\angle A'$: True. Rotation preserves angle measures, so corresponding angles are congruent.
4. "The distance between $P$ and $A$ equals the difference between $P$ and $B'$": False. Rotation gives $PA = PA'$, and $PA$ is not equal to the difference of $P$ and $B'$.
5. $m\angle B = m\angle C'$: False. $\angle B$ corresponds to $\angle B'$, not $\angle C'$, so their measures are not equal.
6. "None of the above": False, as one statement is true.

Answer:

• $m\angle A = m\angle A'$