Problem 208

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Problem 208: (2013 AMC 10b, 23; 12b 19)

In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD} \perp \overline{BC}$, $\overline{DE} \perp \overline{AC}$, and $\overline{AF} \perp \overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $\textbf{(A)}\ 18\qquad$ $\textbf{(B)}\ 21\qquad$ $\textbf{(C)}\ 24\qquad$ $\textbf{(D)}\ 27\qquad$ $\textbf{(E)}\ 30$

Solution

Exercise 208: Solution

In triangle $ABC$, $AB=13$, $BC=21$, and $CA=20$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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