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 are rigid transformations, so they preserve side lengths, distances from the center of rotation, and angle measures.

Step2: Evaluate each statement

1. "The distance between $P$ and $B$ equals the difference between $P$ and $B'$": Rotation preserves distance from $P$, so $PB = PB'$, not the difference. False.
2. $m\angle A = m\angle B'$: Rotation preserves the measure of each original angle, so $\angle A$ maps to $\angle A'$, not $\angle B'$. False.
3. $m\angle A = m\angle A'$: Rotation preserves angle measures, so the measure of $\angle A$ equals the measure of its image $\angle A'$. True.
4. "The distance between $P$ and $A$ equals the difference between $P$ and $B'$": Rotation preserves $PA = PA'$ and $PB = PB'$, but $PA

eq |PB' - PA|$. False.

1. $m\angle B = m\angle C'$: Rotation maps $\angle B$ to $\angle B'$, not $\angle C'$, so their measures are not equal. False.

Answer:

• $m\angle A = m\angle A'$