Problem 105

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Problem 105: (2018 AMC 10a, 24; 2018 AMC12a 18)

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $\textbf{(A) }60 \qquad$ $ \textbf{(B) }65 \qquad$ $ \textbf{(C) }70 \qquad$ $ \textbf{(D) }75 \qquad$ $ \textbf{(E) }80 \qquad$

Solution

Exercise 105: Solution

Triangle $ABC$ with $AB=60$ and $AC=20$ has area $160$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?

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