Question

in circle ( o ), ( mangle poq = 60^{circ} ) and the length of ( overparen{pq}=\frac{4}{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 \(s=\frac{n\pi r}{180}\), where \(n\) is the central angle in degrees and \(r\) is the radius of the circle. Here, \(n = 60^{\circ}\), \(s=\frac{4}{3}\pi\), and \(r = OP\).

Step2: Substitute the values into the formula

Substitute \(n = 60\) and \(s=\frac{4}{3}\pi\) into \(s=\frac{n\pi r}{180}\). We get \(\frac{4}{3}\pi=\frac{60\pi\times OP}{180}\).

Step3: Simplify the equation

Simplify the right - hand side of the equation \(\frac{60\pi\times OP}{180}=\frac{\pi\times OP}{3}\). So, the equation becomes \(\frac{4}{3}\pi=\frac{\pi\times OP}{3}\).

Step4: Solve for \(OP\)

Multiply both sides of the equation \(\frac{4}{3}\pi=\frac{\pi\times OP}{3}\) by \(3\) to get \(4\pi=\pi\times OP\). Then divide both sides by \(\pi\).

Answer:

\(OP = 4\)