To find the diameter of the circular window, we can use the information provided. Here is a detailed, step-by-step solution:
1. Understanding the Problem:
- We have a circular window with an [tex]$8$[/tex]-foot horizontal shelf.
- There is a [tex]$2$[/tex]-foot vertical brace that bisects the horizontal shelf and passes through the center of the window.
2. Recognize the Geometry:
- The horizontal shelf runs across the diameter of the circular window.
- The vertical brace runs perpendicular to the shelf and also intersects the center of the circle.
3. Divide and Conquer:
- By dividing the horizontal shelf into two equal parts, we create two right triangles. Each part of the shelf will be [tex]$4$[/tex] feet (since [tex]$8$[/tex] feet divided by [tex]$2$[/tex] is [tex]$4$[/tex] feet).
4. Identify the Elements of the Right Triangle:
- The base of the right triangle is [tex]$4$[/tex] feet (half of the shelf).
- The height of the right triangle is [tex]$2$[/tex] feet (the brace).
5. Use the Pythagorean Theorem:
- The radius of the circular window is the hypotenuse of the right triangle formed by the base (half of the shelf) and the height (the brace).
- The Pythagorean theorem states that in a right triangle, [tex]\( a^2 + b^2 = c^2 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the legs, and [tex]\( c \)[/tex] is the length of the hypotenuse (the radius in our case).
6. Calculate the Radius:
- Let's denote the radius by [tex]\( r \)[/tex].
- According to the Pythagorean theorem:
[tex]\[
r^2 = 4^2 + 2^2
\][/tex]
[tex]\[
r^2 = 16 + 4
\][/tex]
[tex]\[
r^2 = 20
\][/tex]
[tex]\[
r = \sqrt{20} = \sqrt{4 \times 5} = 2 \sqrt{5} \approx 4.47213595499958
\][/tex]
7. Determine the Diameter:
- The diameter of the circle is twice the radius.
- Hence, the diameter [tex]\( D \)[/tex] is given by:
[tex]\[
D = 2 \times r = 2 \times 4.47213595499958 \approx 8.94427190999916
\][/tex]
Therefore, the diameter of the window is approximately [tex]\( 8.944 \)[/tex] feet.
Diamond = 8.944 feet
