Problem 10

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Problem 10: (2016 AMC 12a, 12)

In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?

$\textbf{(A)}\ 3:2\qquad$ $\textbf{(B)}\ 5:3\qquad$ $\textbf{(C)}\ 2:1\qquad$ $\textbf{(D)}\ 7:3\qquad$ $\textbf{(E)}\ 5:2$

Solution

Exercise 10: Solution

In $\triangle ABC$, $AB=9$, $BC=8$, and $CA=10$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?

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