Question

question
in circle f, fg = 2 and the area of shaded sector = 5/3 π. find the length of gih. express
your answer as a fraction times π.

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 \(r = FG=2\) and \(A=\frac{5}{3}\pi\). Substitute into the formula: \(\frac{5}{3}\pi=\frac{1}{2}(2)^{2}\theta\).

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

Simplify the right - hand side of the equation \(\frac{5}{3}\pi=\frac{1}{2}(4)\theta\), which becomes \(\frac{5}{3}\pi = 2\theta\). Then \(\theta=\frac{5\pi}{6}\).

Step3: Recall the formula for the arc - length

The formula for the arc - length \(s\) of a sector is \(s = r\theta\). Since \(r = 2\) and \(\theta=\frac{5\pi}{6}\). Substitute into the formula: \(s=2\times\frac{5\pi}{6}\).

Step4: Simplify the arc - length expression

\(s=\frac{10\pi}{6}=\frac{5\pi}{3}\).

Answer:

\(\frac{5\pi}{3}\)