AMC 12 2025a Problem 16

Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$. What is $BP$?

\textbf{(A)}~18 \textbf{(B)}~19 \textbf{(C)}~20 \textbf{(D)}~21 \textbf{(E)}~22