To determine the diameter of the circular window given the measurements of an 8-foot horizontal shelf and a 2-foot vertical brace, we can use the Pythagorean Theorem. Here’s the step-by-step approach:
1. Understanding the problem:
- The horizontal shelf is 8 feet long and it sits halfway across the window, making its midpoint align with the center of the window.
- The brace is 2 feet high and reaches vertically from the shelf to the top of the frame, passing through the circle's center.
2. Forming a right triangle:
- One leg of the triangle is half of the shelf's length: [tex]\( \frac{8}{2} = 4 \)[/tex] feet.
- The other leg is the height of the brace: 2 feet.
- The hypotenuse of this right triangle is the radius of the window.
3. Applying the Pythagorean Theorem:
- The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (radius) is equal to the sum of the squares of the other two sides (legs).
[tex]\[
(radius)^2 = (4 \, \text{ft})^2 + (2 \, \text{ft})^2
\][/tex]
[tex]\[
(radius)^2 = 16 \, \text{ft}^2 + 4 \, \text{ft}^2
\][/tex]
[tex]\[
(radius)^2 = 20 \, \text{ft}^2
\][/tex]
4. Calculating the radius:
- Taking the square root of both sides to find the radius:
[tex]\[
radius = \sqrt{20} \, \text{ft} \approx 4.472 \, \text{ft}
\][/tex]
5. Determining the diameter:
- The diameter of the window is twice the radius:
[tex]\[
diameter = 2 \times radius
\][/tex]
[tex]\[
diameter \approx 2 \times 4.472 \, \text{ft} \approx 8.944 \, \text{ft}
\][/tex]
So, the diameter of the window is approximately [tex]\( 8.944 \)[/tex] feet.
Therefore, the diameter of the window is [tex]\(\boxed{8.944} \text{ feet}\)[/tex].
