16. In triangle $ABC$, sides $AB$ and $AC$ measure $10 \, \text{m}$ and $17 \, \text{m}$ respectively, the height $AH$ measures $8 \, \text{m}$, and angle $\widehat{A}$ is obtuse. Indicating with $D$ the midpoint of $AB$, with $E$ the midpoint of $BD$, with $F$ the midpoint of $AC$, and with $G$ the midpoint of $AF$, how many $\text{m}^2$ does the area of triangle $DEG$ measure?
(A) $9 / 2$
(B) $19 / 4$
(C) $21 / 4$
(D) $25 / 4$
(E) $11 / 2$
(A) 876
(B) 874
(C) 877
(D) 701
(E) 699
Problem 16. The correct answer is (C).

First, we calculate the area $[A B C]$ of triangle $A B C$. Using the Pythagorean theorem, we find that (in meters) $\overline{B H}=\sqrt{\overline{A B}^{2}-\overline{A H}^{2}}=6 \quad$ and $\overline{C H}=\sqrt{\overline{A C}^{2}-\overline{A H}^{2}}=15$, from which $\overline{B C}=21 \quad$ and $[A B C]=\frac{21 \cdot 8}{2}=84 \mathrm{~m}^{2}$. The area sought $[D E G]$ is therefore (in $\mathrm{m}^{2}$ ):

$$
[D E G]=\frac{1}{3} \cdot[A E G]=\frac{1}{3} \cdot\left(\frac{3}{4} \cdot[A B G]\right)=\frac{1}{3} \cdot\left(\frac{3}{4} \cdot\left(\frac{1}{4} \cdot[A B C]\right)\right)=\frac{1}{16} \cdot[A B C]=\frac{21}{4}
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_3444f6e2e195aabb18c2g-3.jpg?height=290&width=651&top_left_y=2148&top_left_x=756)