Question

in circle c, a central angle of 60° intercepts arc pqr as shown in the figure. if arc pqr has a length of π centimeters (cm), what is the value of x?

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{\theta}{360^{\circ}}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle. In this case, \(\theta = 60^{\circ}\), \(s=\pi\) cm, and \(r = x\) cm.

Step2: Substitute the values into the formula

Substitute \(\theta = 60^{\circ}\) and \(s=\pi\) into \(s=\frac{\theta}{360^{\circ}}\times2\pi r\). We get \(\pi=\frac{60^{\circ}}{360^{\circ}}\times2\pi x\).
Simplify \(\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}\). So the equation becomes \(\pi=\frac{1}{6}\times2\pi x\).

Step3: Solve for \(x\)

First, simplify the right - hand side of the equation: \(\frac{1}{6}\times2\pi x=\frac{\pi x}{3}\).
Now we have the equation \(\pi=\frac{\pi x}{3}\).
Multiply both sides of the equation by \(3\) to get \(3\pi=\pi x\).
Divide both sides by \(\pi\). Since \(\pi
eq0\), we have \(x = 3\).

Answer:

\(3\)