Question

custom drapes are being fitted for a large circular window. it is difficult to get these measurements, but the window has an 8 ft horizontal shelf with a 2 ft 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 = \boxed{ } feet

Explanation:

Step1: Identify the problem type

This is a problem related to circles, specifically using the Pythagorean theorem in a circle context (chord and radius/diameter relationship). Let the radius of the circle be \( r \) feet. The horizontal shelf length is 8 ft, so half of it is \( \frac{8}{2}=4 \) ft. The brace length is 2 ft, so the distance from the center of the circle to the shelf (chord) is \( r - 2 \) ft.

Step2: Apply the Pythagorean theorem

For a circle, the relationship between the radius \( r \), half of the chord length (\( a \)), and the distance from the center to the chord (\( d \)) is given by \( r^{2}=a^{2}+d^{2} \). Here, \( a = 4 \) and \( d=r - 2 \). So we have the equation:
\[
r^{2}=4^{2}+(r - 2)^{2}
\]

Step3: Expand and solve the equation

Expand \( (r - 2)^{2} \) using the formula \( (x - y)^{2}=x^{2}-2xy + y^{2} \). So \( (r - 2)^{2}=r^{2}-4r + 4 \). Substitute into the equation:
\[
r^{2}=16+r^{2}-4r + 4
\]
Subtract \( r^{2} \) from both sides:
\[
0=20 - 4r
\]
Add \( 4r \) to both sides:
\[
4r=20
\]
Divide both sides by 4:
\[
r = 5
\]

Step4: Find the diameter

The diameter \( d \) of a circle is twice the radius, so \( d = 2r \). Substitute \( r = 5 \):
\[
d=2\times5 = 10
\]

Answer:

10