The area of square $A B C D$ is 64 . The midpoints of its sides are joined to form the square $E F G H$. The midpoints of its sides are $J, K, L$, and $M$. The area of the shaded region is
(A) 32
(D) 28
(B) 24
(E) 16
(C) 20

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-139.jpg?height=336&width=334&top_left_y=2055&top_left_x=1340)
The area of square $A B C D$ is 64 . The midpoints of its sides are joined to form the square $E F G H$. The midpoints of its sides are $J, K, L$, and $M$. The shaded area is
(A) 32
(B) 24
(C) 20
(D) 28
(E) 16

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-233.jpg?height=406&width=412&top_left_y=342&top_left_x=1409)

## Solution

If we join vertices as illustrated in the diagram, we note that the square $F G H E$ is divided into eight equal parts, six of which are shaded. Since the square $F G H E$ is itself one half the area of the larger square $A B C D$, the shaded region is then, $\frac{6}{8} \times\left(\frac{1}{2} \times 64\right)=24$.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-233.jpg?height=466&width=496&top_left_y=843&top_left_x=1411)