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Grey square A C D E is shown with white square A B F G in the upper left. A 4 inch line segment extends from point G to point D.
ABGF is a square with half the perimeter of square ACDE. GD=4 in.
. Find the area of the shaded region.
(3 points)
First, let's find the perimeter of square ACDE. Since ABGF has half the perimeter of ACDE, we can find the perimeter of ABGF first. Let's say the side length of ABGF is x inches. So, the perimeter of ABGF is 4x.
Since ABGF has half the perimeter of ACDE, the perimeter of ACDE is 8x.
Since all sides of a square are equal, each side of ACDE is 8x/4 = 2x inches.
Now, we know that GD is 4 inches. Since GD is the diagonal of the square ACDE and we have the length of one side (2x inches), we can use the Pythagorean theorem to find x.
(2x)^2 + (2x)^2 = 4^2
4x^2 + 4x^2 = 16
8x^2 = 16
x^2 = 2
x = √2
Now, we can find the area of the shaded region which is the area of square ACDE minus the area of square ABGF.
Area of ACDE = (2x)^2 = 2^2 = 4 square inches
Area of ABGF = x^2 = (√2)^2 = 2 square inches
Area of shaded region = 4 - 2 = 2 square inches
So, the area of the shaded region is 2 square inches.