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 = feet

Explanation:

Step1: Find half - chord length

The shelf is a chord of the circle. Its length is 8 ft. The perpendicular from the center of the circle to the chord bisects the chord. So the half - chord length $a = \frac{8}{2}=4$ ft.

Step2: Identify the distance from center to chord

The length of the brace is 2 ft. Let the radius of the circle be $r$. The distance from the center of the circle to the chord is $r - 2$.

Step3: Apply the Pythagorean theorem

In the right - triangle formed by the radius, half - chord, and the line from the center to the mid - point of the chord, we have $a^{2}+(r - 2)^{2}=r^{2}$ according to the Pythagorean theorem. Substitute $a = 4$ into the equation: $4^{2}+(r - 2)^{2}=r^{2}$. Expand the equation: $16+r^{2}-4r + 4=r^{2}$. Simplify the equation: $16+4-4r=0$, $20 - 4r=0$, $4r = 20$, $r = 5$ ft.

Step4: Calculate the diameter

The diameter $d$ of a circle is $d = 2r$. Since $r = 5$ ft, $d=10$ ft.

Answer:

10