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.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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)
1 answer