ABGF is a square with half the perimeter of square ACDE. GD = 4in. Find the area of the shaded region.

Please help me work this problem out I am stuck.
It's a square and in the top right corner there in another square inside it that is 1/4 the size of the big one. From that corner there is a diagonal line going to the other corner saying GD=4in
The shaded area is the 3/4 that are not part of the little 1/4 square
Please help me

Someone told me that the answer is 24, which is correct, but I'm not sure how they got it :(, but hopefully that at least helps a little.

To solve this problem, we need to follow a step-by-step process:

Step 1: Understand the problem and draw a diagram.
From the problem description, we have two squares: ABGF (big square) and ACDE (bigger square). The big square ABGF has half the perimeter of the bigger square ACDE. We are given that GD = 4 inches, and we need to find the area of the shaded region.

Step 2: Label the diagram and use given information to find missing dimensions.
Let's assume the side length of the bigger square ACDE is "x". Since the big square ABGF has half the perimeter of the bigger square ACDE, its side length will be x/2. We are also told that GD = 4 inches.

Step 3: Use the labeled diagram to find the area of the shaded region.
The shaded region is the part of the big square ABGF that is not covered by the smaller square. To find the area of the shaded region, we need to subtract the area of the smaller square from the area of the big square.

Area of the big square ABGF = (side length)^2
= (x/2)^2
= x^2/4

Area of the smaller square = (side length)^2
= (x/4)^2
= x^2/16

Area of the shaded region = Area of the big square - Area of the smaller square
= x^2/4 - x^2/16
= (4x^2 - x^2)/16
= 3x^2/16

Therefore, the area of the shaded region is 3x^2/16 square units.

Step 4: Substitute the given value and solve for the area of the shaded region.
From the problem, we are given GD = 4 inches. Since GD is the diagonal of the smaller square, we can use the Pythagorean theorem to find x.

By Pythagorean theorem,
(x/4)^2 + (x/4)^2 = GD^2
x^2/16 + x^2/16 = 4^2
2x^2/16 = 16
x^2/8 = 16
x^2 = 128
x = √128
x ≈ 11.31 inches

Now that we have the value of x, we can substitute it into our earlier equation to find the area of the shaded region.

Area of the shaded region = 3x^2/16
= 3(11.31)^2/16
≈ 23.99 square units

Therefore, the area of the shaded region is approximately 23.99 square units.

To find the area of the shaded region, we need to first find the dimensions of the squares ACDE and ABGF.

Given that GD = 4 inches, we can see that GD is the diagonal of the smaller square ABGF. Since GD is the hypotenuse of a right-angled triangle, we can use the Pythagorean theorem to find the length of the sides of the smaller square.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (GD) is equal to the sum of the squares of the lengths of the other two sides.

Let's say the side length of the smaller square is x inches. Using the Pythagorean theorem, we have:

x^2 + x^2 = GD^2
2x^2 = GD^2
2x^2 = 4^2
2x^2 = 16
x^2 = 16/2
x^2 = 8
x = √8
x ≈ 2.83 inches

So, the side length of the smaller square ABGF is approximately 2.83 inches.

Now, we know that the perimeter of the larger square ACDE is twice the sum of the side lengths of the smaller square, since ABGF is 1/4 the size of ACDE.

Perimeter of ACDE = 2 * (AB + BC + CD + DE)
Perimeter of ACDE = 2 * (2.83 + 2.83 + 2.83 + 2.83)
Perimeter of ACDE = 2 * (4 * 2.83)
Perimeter of ACDE ≈ 22.64 inches

Since ABGF is a square with half the perimeter of ACDE, its perimeter can be found by dividing the perimeter of ACDE by 2.

Perimeter of ABGF = 22.64/2
Perimeter of ABGF ≈ 11.32 inches

Since ABGF is a square, all of its sides are equal. So, the length of each side of ABGF can be found by dividing its perimeter by 4.

Length of each side of ABGF = 11.32/4
Length of each side of ABGF ≈ 2.83 inches

The area of the shaded region is equal to the area of the larger square ACDE minus the area of the smaller square ABGF.

Area of the shaded region = Area of ACDE - Area of ABGF
Area of the shaded region = (side length of ACDE)^2 - (side length of ABGF)^2
Area of the shaded region = (2.83)^2 - (2.83)^2
Area of the shaded region ≈ 8 - 8
Area of the shaded region ≈ 0 square inches

Therefore, the area of the shaded region is approximately 0 square inches.