Question

consider circle c with angle acb measuring $\frac{3\pi}{4}$ radians. if minor arc ab measures $9\pi$ inches, what is the length of the radius of circle c? if necessary, round your answer to the nearest inch. 6 inches 12 inches 18 inches 24 inches

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step2: Substitute the given values into the formula

We are given that \(s=9\pi\) inches and \(\theta = \frac{3\pi}{4}\) radians. Substituting into \(s = r\theta\), we get \(9\pi=r\times\frac{3\pi}{4}\).

Step3: Solve for \(r\)

To solve for \(r\), we can first cancel out \(\pi\) from both sides of the equation \(9\pi=r\times\frac{3\pi}{4}\). We get \(9 = r\times\frac{3}{4}\). Then, multiply both sides by \(\frac{4}{3}\) to isolate \(r\). So \(r=\frac{9\times4}{3}\).

$$r = 12$$

Answer:

12 inches