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]
