Make this "Given that GD = 5 inches, we can find the side length of square ABGF:
1. **Side Length of ABGF**:
- The perimeter of ABGF is half of the perimeter of ACDE.
- Let the side length of ABGF be \(a\). Then the side length of ACDE is \(2a\).
- The perimeter of ACDE is \(4 \times 2a = 8a\).
- Since ABGF has half the perimeter, we have: \(8a = 5\).
- Solving for \(a\): \(a = \frac{5}{8} = \frac{5}{8}\) inches.
2. **Area of Square ABGF**:
- The area of a square is given by \(A_{\text{square}} = (\text{side length})^2\).
- So, the area of square ABGF is: \(A_{\text{ABGF}} = a^2 = \left(\frac{5}{8}\right)^2 = \frac{25}{64}\) square inches.
3. **Area of Square GDHF**:
- Since GDHF is also a square with the same side length, its area is the same as that of ABGF:
\(A_{\text{GDHF}} = A_{\text{ABGF}} = \frac{25}{64}\) square inches.
4. **Area of Shaded Region**:
- The shaded region consists of two squares (ABGF and GDHF) and two right triangles (AGD and BFH).
- The area of the shaded region is the difference between the area of square ACDE and the sum of the areas of ABGF and GDHF:
\[A_{\text{shaded}} = A_{\text{ACDE}} - (A_{\text{ABGF}} + A_{\text{GDHF}})\]
- The area of square ACDE is \((2a)^2 = 4a^2 = 4\left(\frac{5}{8}\right)^2 = \frac{100}{64}\) square inches.
- Subtracting the areas:
\[A_{\text{shaded}} = \frac{100}{64} - 2 \cdot \frac{25}{64} = \frac{50}{64} = \frac{25}{32}\] square inches.
5. **Rounded Answer**:
- Rounding to the nearest tenth: \(A_{\text{shaded}} \approx 0.8\) square inches.
Therefore, the approximate area of the shaded region is **0.8 square inches**. 🌟📏🔲🔴
For more precise calculations, you can use the exact value of the square root of 2." into a short answer showing all work in order
Given that GD = 5 inches, we can find the side length of square ABGF:
1. **Side Length of ABGF**:
- Perimeter of ACDE = 8a (where a is the side length of ABGF)
- Perimeter of ABGF = 4a
- Since ABGF has half the perimeter of ACDE, 8a = 5
- Solving for a: a = 5/8 inches
2. **Area of Square ABGF**:
- Area of square ABGF = (5/8)^2 = 25/64 square inches
3. **Area of Square GDHF**:
- Area of square GDHF = same as area of ABGF = 25/64 square inches
4. **Area of Shaded Region**:
- Area of shaded region = Area of square ACDE - (Area of ABGF + Area of GDHF)
- Area of square ACDE = 100/64 square inches
- Area of shaded region = 100/64 - 2 * 25/64 = 50/64 = 25/32 square inches
5. **Rounded Answer**:
- Approximate area of shaded region is 0.8 square inches
Therefore, the approximate area of the shaded region is **0.8 square inches**. 🌟📏🔲🔴
For more precise calculations, the exact value of the square root of 2 can be used.