To determine the diameter of the circular window, we can use some properties of circles and right triangles. Here's a step-by-step approach to solve the problem:
1. Identify the Known Measurements:
- The "horizontal shelf" is 8 feet long. This shelf acts as a chord of the circle.
- The "brace" is 2 feet long. This brace reaches from the midpoint of the chord to the center of the circle and is perpendicular to the chord.
2. Visualize the Geometry:
- Let's place the chord (shelf) horizontally with its midpoint directly below the center of the circle.
- The brace is a vertical line from the midpoint of the chord to the center.
3. Form a Right Triangle:
- The brace acts as one leg of the right triangle. Its length is 2 feet.
- Half of the chord's length is another leg of the right triangle. The length of half the chord is \( \frac{8}{2} = 4 \) feet.
- The radius of the circle is the hypotenuse of this right triangle.
4. Apply the Pythagorean Theorem:
- According to the Pythagorean theorem, in a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the relationship is \( a^2 + b^2 = c^2 \).
- Here, \( a = 2 \) feet (brace), \( b = 4 \) feet (half-chord), and \( c \) (radius) is what we are trying to find.
5. Calculate the Radius:
- Substitute the known values into the Pythagorean Theorem equation:
[tex]\[
2^2 + 4^2 = \text{radius}^2
\][/tex]
[tex]\[
4 + 16 = \text{radius}^2
\][/tex]
[tex]\[
\text{radius}^2 = 20
\][/tex]
[tex]\[
\text{radius} = \sqrt{20} \approx 4.472
\][/tex]
6. Calculate the Diameter:
- The diameter of the circle is twice the radius.
- Therefore, the diameter \( \text{diameter} = 2 \times 4.472 \).
7. Final Answer:
- The diameter of the window is approximately \( \boxed{8.944} \) feet.
Thus, the diameter of the circular window is approximately 8.944 feet.
