Question

in circle q, the length of \\( \overparen{r s}=3 \pi \\) and \\( m \angle r q s=90^{circ} \\). find the area shaded below. express your answer as a fraction times \\( \pi \\).

Explanation:

Step1: Find the radius of the circle

The formula for the length of an arc is \(L=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle and \(r\) is the radius.
Given \(L = 3\pi\) and \(\theta=90^{\circ}\), substitute into the formula:
\(3\pi=\frac{90}{360}\times2\pi r\)
Simplify \(\frac{90}{360}=\frac{1}{4}\), so \(3\pi=\frac{1}{4}\times2\pi r\)
\(3\pi=\frac{\pi r}{2}\)
Multiply both sides by \(2\): \(6\pi=\pi r\), then \(r = 6\)

Step2: Find the area of the shaded region

The area of a circle is \(A=\pi r^{2}\). The central angle of the shaded region is \(360 - 90=270^{\circ}\)
The formula for the area of a sector is \(A=\frac{\theta}{360}\times\pi r^{2}\)
Substitute \(\theta = 270^{\circ}\) and \(r = 6\)
\(A=\frac{270}{360}\times\pi\times6^{2}\)
Simplify \(\frac{270}{360}=\frac{3}{4}\) and \(6^{2}=36\)
\(A=\frac{3}{4}\times\pi\times36\)
\(A = 27\pi\)

Answer:

\(27\pi\)