ABGF is a square with half the perimeter of square ACDE. GD=5 in. Find the area of the shaded region. Round your answer to the nearest tenth. Please show all work in order to receive full credit. Either type it out or you may use the l symbol to upload a picture of your work.
First, let's find the perimeter of square ACDE.
Let x be the side length of square ACDE.
Perimeter of square ACDE = 4x
Perimeter of square ABGF = 2x
Given that the perimeter of square ACDE is twice the perimeter of square ABGF:
4x = 2(2x)
4x = 4x
Since x = x, this means that square ACDE and square ABGF are the same size.
Since GD=5 in, then each side of square ACDE is 10 in (5 in for GD and 5 in for DF).
Therefore, the area of square ACDE is:
Area = side length^2 = 10^2 = 100 sq in.
Since ABGF is the same size as ACDE, the area of square ABGF is also 100 sq in.
Now, let's find the area of the shaded region, which consists of two quarter circles.
Since the radius of the quarter circle is the same as the side length of square ABGF (5 in), the area of each quarter circle is:
Area of one quarter circle = (1/4) * π * r^2 = (1/4) * π * 5^2 = (1/4) * π * 25 ≈ 19.63 sq in.
Since there are two quarter circles in the shaded region, the total area of the shaded region is approximately:
2 * 19.63 ≈ 39.26 sq in.
Therefore, the area of the shaded region is approximately 39.3 sq in.