Custom drapes are being fitted for a large circular window. It is difficult to get these measurements, but the window has an [tex]$8 \, \text{ft}[tex]$[/tex] horizontal shelf with a [tex]$[/tex]2 \, \text{ft}$[/tex] 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 find the diameter of the circular window, follow these steps:

1. Understand the Problem:
- The window has an 8-foot horizontal shelf that acts as the base of a right triangle when it’s bisected.
- A 2-foot brace runs from the base to the top of the window, serving as the height of the right triangle.
- The top point where the brace touches the window is the center of the circle.

2. Identify the Components:
- The base of the right triangle is half of the shelf, which is 8 feet / 2 = 4 feet.
- The height of the right triangle is the length of the brace, which is 2 feet.

3. Apply the Pythagorean Theorem:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
- Here, the hypotenuse represents the radius \(r\) of the circle.
- Hence, we have:
[tex]\[ r = \sqrt{(4^2 + 2^2)} \][/tex]

4. Calculate the Radius:
- \( 4^2 = 16 \)
- \( 2^2 = 4 \)
- Adding these together, we get:
[tex]\[ 16 + 4 = 20 \][/tex]
- Taking the square root of 20:
[tex]\[ r = \sqrt{20} \approx 4.47213595499958 \][/tex]

5. Determine the Diameter:
- The diameter \(D\) of the circle is twice the radius:
[tex]\[ D = 2 \times r = 2 \times 4.47213595499958 \approx 8.94427190999916 \][/tex]

Thus, the diameter of the window is approximately \(8.944\) feet. Therefore:
[tex]\[ \text{Diameter} = 8.94427190999916 \text{ feet} \][/tex]