Custom drapes are being fitted for a large circular window. The window has an 8 ft horizontal shelf with a 2 ft brace that sits in the frame. If the brace is extended upward, it would go through the center of the shelf and the circle. What is the diameter of the window?

Diameter [tex] = \square \text{ feet}[/tex]



Answer :

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].