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.
First, we need to find the perimeter of square ACDE. Since ABGF is a square with half the perimeter of square ACDE, we can say that the perimeter of ABGF is half of the perimeter of ACDE.
Let the side length of square ACDE be x.
Perimeter of ACDE = 4x
Perimeter of ABGF = 2x
Given that GD = 5 in, we can find the side length of square ABGF:
2x - 5 = x
x = 5 in
Therefore, the side length of square ABGF is 5 in.
Now, we need to find the area of the shaded region. The shaded region consists of two squares (ABGF and GDHF) and two right triangles (AGD and BFH).
Area of square ABGF = (side length)^2 = (5)^2 = 25 sq in
Area of square GDHF = (side length)^2 = (5)^2 = 25 sq in
The area of the two right triangles can be found using the Pythagorean theorem:
AG^2 + GD^2 = AD^2
AG^2 + 5^2 = 5^2
AG^2 = 0
This means AG = 0, so the area of triangle AGD is 0 sq in.
BF = AG = 0 (since AG = 0), so the area of triangle BFH is also 0 sq in.
Now, add up the areas of the squares and triangles to find the total area of the shaded region:
Total area = 25 + 25 + 0 + 0 = 50 sq in
Therefore, the area of the shaded region is 50 sq in.