Question

ABGF is a square with half the perimeter of square ACDE. GD = 4 in.. Find the area of the shaded region. (image attached)thank you ! :)

Answer 1

The perimeter of ABGF is half that of ACDE; then,

`\begin{gathered} \Rightarrow2(4AB)=4AC \\ \Rightarrow AB=(AC)/(2) \end{gathered}`

Therefore, each side of ABGF is half the length of any side of ACDE.

Then, point G is the middle point of the square ACDE.

Hence, the diagonal AD is 2*4in=8in in length, and with that information, we can calculate the length of a side of ACDE as shown below

Using the Pythagorean theorem,

`\begin{gathered} 8^2=l^2+l^2=2l^2 \\ \Rightarrow l^2=(64)/(2)=32 \\ \Rightarrow l=√(32)=4√(2) \end{gathered}`

Therefore, the area of the square ACDE is (4sqrt2)^2=32 in^2.

Similarly, finding the area of the square ABGF,

`\begin{gathered} 4^2=s^2+s^2=2s^2 \\ \Rightarrow s=√(8) \end{gathered}`

Therefore, the area of square ABGF is s^2=8.

Finally, the area of the shaded region is

`A_(shaded)=32-8=24`

The area of the shaded region is 24 in^2