Question

in circle ( o ), ( op = 4 ) and the area of shaded sector ( =\frac{16}{9}pi ). find the length of ( overparen{pq} ). express your answer as a fraction times ( pi ).

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{1}{2}r^{2}\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central - angle in radians. Given \(r = OP=4\) and \(A=\frac{16}{9}\pi\). Substitute these values into the formula: \(\frac{16}{9}\pi=\frac{1}{2}(4)^{2}\theta\).

Step2: Solve for the central - angle \(\theta\)

First, simplify the right - hand side of the equation \(\frac{16}{9}\pi=\frac{1}{2}\times16\theta\). Then \(\frac{16}{9}\pi = 8\theta\). Solving for \(\theta\), we get \(\theta=\frac{2}{9}\pi\).

Step3: Recall the formula for the arc - length

The formula for the arc - length \(s\) of a circle is \(s = r\theta\). Since \(r = 4\) and \(\theta=\frac{2}{9}\pi\). Substitute these values into the formula: \(s=4\times\frac{2}{9}\pi\).

Answer:

\(\frac{8}{9}\pi\)