Rectangle $A B C D$ has length 9 and width 5. Diagonal $A C$ is divided into 5 equal parts at $W, X, Y$, and $Z$. Determine the area of the shaded region.
(A) 36
(B) $\frac{36}{5}$
(C) 18
(D) $\frac{4 \sqrt{106}}{5}$
(E) $\frac{2 \sqrt{106}}{5}$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-149.jpg?height=293&width=477&top_left_y=1485&top_left_x=1301)
Rectangle $A B C D$ has length 9 and width 5. Diagonal $A C$ is divided into 5 equal parts at $W, X, Y$, and $Z$. Determine the area of the shaded region.
(A) 36
(B) $\frac{36}{5}$
(C) 18
(D) $\frac{4 \sqrt{106}}{5}$
(E) $\frac{2 \sqrt{106}}{5}$

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-281.jpg?height=320&width=523&top_left_y=1385&top_left_x=1365)

## Solution

Triangle $A B C$ has area $\frac{1}{2}(9)(5)=\frac{45}{2}$. Triangles $A B W, W B X, X B Y, Y B Z$, and $Z B C$ have equal bases and altitudes, so the area of each of these small triangles is $\frac{1}{5}\left(\frac{45}{2}\right)=\frac{9}{2}$. Similarly, triangles $A D W$, $W D X, X D Y, Y D Z$, and $Z D C$ each have area $\frac{9}{2}$.

Thus, the shaded region has area $4\left(\frac{9}{2}\right)=18$.

ANSWER: (C)