To determine the diameter of the circular window, we will use the information about the horizontal shelf and the brace, which help us form a right-angled triangle. Here’s a step-by-step explanation:
1. Identify the given components:
- The length of the horizontal shelf is 8 feet.
- The length of the brace is 2 feet.
2. Understand the geometry formed:
- The horizontal shelf and the brace form a right-angled triangle with the center of the circle.
- To simplify, the horizontal shelf can be considered divided into two equal parts by the brace which goes through the center, making each part have a length of [tex]\( 4 \)[/tex] feet.
3. Recognize the right-angled triangle:
- In the right-angled triangle, one leg (half of the horizontal shelf) is 4 feet.
- The other leg (the brace) is 2 feet.
- The hypotenuse of this triangle is the radius of the circle.
4. Apply the Pythagorean theorem:
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ([tex]\( c \)[/tex]) is equal to the sum of the squares of the other two sides ([tex]\( a \)[/tex] and [tex]\( b \)[/tex]):
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Where [tex]\( a = 4 \)[/tex] feet and [tex]\( b = 2 \)[/tex] feet.
5. Calculate the radius of the circle ([tex]\( c \)[/tex]):
[tex]\[
4^2 + 2^2 = c^2
\][/tex]
[tex]\[
16 + 4 = c^2
\][/tex]
[tex]\[
c^2 = 20
\][/tex]
[tex]\[
c = \sqrt{20} \approx 4.47213595499958 \text{ feet}
\][/tex]
6. Determine the diameter of the window:
The diameter ([tex]\( d \)[/tex]) is twice the radius:
[tex]\[
d = 2 \times c = 2 \times 4.47213595499958 \approx 8.94427190999916 \text{ feet}
\][/tex]
So, the diameter of the window is approximately:
[tex]\[
\boxed{8.94427190999916} \text{ feet}
\][/tex]
