Problem 23

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Problem 23: (2016 AMC 12b, 17)

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?

$\textbf{(A)}\ 1 \qquad$ $ \textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad$ $ \textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad$ $ \textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad$ $ \textbf{(E)}\ \frac{6}{5}$

Solution

Exercise 23: Solution

In $\triangle ABC$, $AB=13$, $BC=14$, $CA=15$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?

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