In the diagram, $D E F G$ is a square and $A B C D$ is a rectangle. A straight line is drawn from $A$, passes through $C$ and meets $F G$ at $H$. The area of the shaded region is
(A) 8
(B) 8.5
(C) 10
(D) 9
(E) 10.5

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-149.jpg?height=328&width=423&top_left_y=2232&top_left_x=1342)
In the diagram, $D E F G$ is a square and $A B C D$ is a rectangle. A straight line is drawn from $A$, passes through $C$ and meets $F G$ at $H$. The area of the shaded region is
(A) 8
(B) 8.5
(C) 10
(D) 9
(E) 10.5

## Solution

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-255.jpg?height=352&width=463&top_left_y=214&top_left_x=1400)

We are given that $A H$ is a straight line segment, and $C$ is a point on $A H$. Since $A D: D C=2: 1$, then $A G: G H=2: 1$. Since the length of $A G$ is 6 , the length of $G H$ is 3 .

The area of rectangle $A B C D$ is $1 \times 2=2$. The area of square $D E F G$ is $4^{2}=16$. The area of $\triangle A H G$ is $\frac{1}{2}(6)(3)=9$.

Therefore, the area of the shaded region is $2+16-9=9$.

ANSWER: (D)