Abgf is a square with half the perimeter of square acde. gd = 4in. find the area of the shaded region.

The complete question in the attached figure

Let
x---------> length side square ACDE

the perimeter of square ACDE=x+x+x+x--------> 4x

we know that
ABGF is a square with half the perimeter of square ACDE

then
perimeter of square ABGF=4x/2-------> 2x
the length side of square ABGF is 2x/4------> 0.50x

the area of the shaded region=[area square ACDE]-[area square ABGF]
 the area of the shaded region=[x*x]-[0.50x*0.50x]---> x²-0.25x²---> 0.75x²
the area of the shaded region= 0.75x²

find the value of x
applying the Pythagorean theorem

4²=(0.5x)²+(0.5x)²------> 16=0.25x²+0.25x²------> 16=0.50x²
x²=32-------> x=4√2 in

the area of the shaded region= 0.75x²----> 0.75*(4√2)²---> 0.75*32=24 in²

the answer is
the area of the shaded region is 24 in²

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