Question

in circle ( o ), ( mangle poq = 60^{circ} ) and the length of ( overparen{pq}=\frac{2}{3}pi ). find the length of ( overline{op} ).

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{n}{360}\times2\pi r\), where \(n\) is the central angle in degrees and \(r\) is the radius of the circle. Here, \(n = 60^{\circ}\), \(s=\frac{2}{3}\pi\), and \(r = OP\).

Step2: Substitute the values into the formula

Substitute \(n = 60\) and \(s=\frac{2}{3}\pi\) into \(s=\frac{n}{360}\times2\pi r\). We get \(\frac{2}{3}\pi=\frac{60}{360}\times2\pi r\).
Simplify \(\frac{60}{360}=\frac{1}{6}\). So the equation becomes \(\frac{2}{3}\pi=\frac{1}{6}\times2\pi r\).
Simplify the right - hand side: \(\frac{1}{6}\times2\pi r=\frac{1}{3}\pi r\).

Step3: Solve for \(r\)

We have the equation \(\frac{2}{3}\pi=\frac{1}{3}\pi r\).
Multiply both sides of the equation by \(3\) to get \(2\pi=\pi r\).
Divide both sides by \(\pi\). We find that \(r = 2\).

Answer:

\(2\)