A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral? $\mathrm{(A) \ } 15\qquad$ $ \mathrm{(B) \ } 17\qquad$ $ \mathrm{(C) \ } \frac{35}{2}\qquad$ $ \mathrm{(D) \ } 18\qquad$ $ \mathrm{(E) \ } \frac{55}{3}$
Solution
Connect $CD$.
Since $D$ is the midpoint of $AE$, $3+x=y$, $x:(y+7)=3:7$, or $7x=3y+21$.
In $\triangle ABC$, point $E$ lies on $\overline{BC}$, point $F$ lies on $\overline{AC}$, and segments $\overline{AE}$ and $\overline{BF}$ intersect at $D$. The areas of $\triangle ADF$, $\triangle ADB$, and $\triangle BDE$ are $3$, $5$, and $6$, respectively. What is the area of quadrilateral $DECF$?
$\mathrm{(A) \ } 15\qquad$ $ \mathrm{(B) \ } 17\qquad$ $ \mathrm{(C) \ } \frac{35}{2}\qquad$ $ \mathrm{(D) \ } 18\qquad$ $ \mathrm{(E) \ } \frac{55}{3}$ 
