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# Difference between revisions of "2018 AMC 10A Problems/Problem 24"

## Problem

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 1

Let $BC = a$, $BG = x$, $GC = y$, and the length of the perpendicular to $BC$ through $A$ be $h$. By angle bisector theorem, we have that $$\frac{50}{x} = \frac{10}{y},$$ where $y = -x+a$. Therefore substituting we have that $BG=\frac{5a}{6}$. By similar triangles, we have that $DF=\frac{5a}{12}$, and the height of this trapezoid is $\frac{h}{2}$. Then, we have that $\frac{ah}{2}=120$. We wish to compute $\frac{5a}{8}\cdot\frac{h}{2}$, and we have that it is $\boxed{75}$ by substituting. (rachanamadhu)

## Solution 2

For this problem, we have $\triangle{ADE}\sim\triangle{ABC}$ because of SAS and $DE = \frac{BC}{2}$. Therefore, $\bigtriangleup ADE$ is a quarter of the area of $\bigtriangleup ABC$, which is $30$. Subsequently, we can compute the area of quadrilateral $BDEC$ to be $120 - 30 = 90$. Using the angle bisector theorem in the same fashion as the previous problem, we get that $\overline{BG}$ is $5$ times the length of $\overline{GC}$. We want the larger piece, as described by the problem. Because the heights are identical, one area is $5$ times the other, and $\frac{5}{6} \cdot 90 = \boxed{75}$.

## Solution 3

The area of $\bigtriangleup ABG$ to the area of $\bigtriangleup ACG$ is $5:1$ by Law of Sines. So the area of $\bigtriangleup ABG$ is $100$. Since $\overline{DE}$ is the midsegment of $\bigtriangleup ABC$, so $\overline{DF}$ is the midsegment of $\bigtriangleup ABG$ . So the area of $\bigtriangleup ACG$ to the area of $\bigtriangleup ABG$ is $1:4$ , so the area of $\bigtriangleup ACG$ is $25$, by similar triangles. Therefore the area of quad $FDBG$ is $100-25=\boxed{75}$

## Solution 4

The area of quadrilateral $FDBG$ is the area of $\bigtriangleup ABG$ minus the area of $\bigtriangleup ADF$. Notice, $\overline{DE} || \overline{BC}$, so $\bigtriangleup ABG \sim \bigtriangleup ADF$, and since $\overline{AD}:\overline{AB}=1:2$, the area of $\bigtriangleup ADF:\bigtriangleup ABG=(1:2)^2=1:4$. Given that the area of $\bigtriangleup ABC$ is $120$, using $\frac{bh}{2}$ on side $AB$ yields $\frac{50h}{2}=120\implies h=\frac{240}{50}=\frac{24}{5}$. Using the Angle Bisector Theorem, $\overline{BG}:\overline{BC}=50:(10+50)=5:6$, so the height of $\bigtriangleup ABG: \bigtriangleup ACB=5:6$. Therefore our answer is $\big[ FDBG\big] = \big[ABG\big]-\big[ ADF\big] = \big[ ABG\big]\big(1-\frac{1}{4}\big)=\frac{3}{4}\cdot \frac{bh}{2}=\frac{3}{8}\cdot 50\cdot \frac{5}{6}\cdot \frac{24}{5}=\frac{3}{8}\cdot 200=\boxed{75}$ -Solution by ktong