Question

in circle ( w ), ( mangle xwy = 20^{circ} ) and the length of ( overparen{xy}=\frac{2}{3}pi ). find the length of ( overline{wx} ).

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, \(n = m\angle XWY=20^{\circ}\), \(s=\overset{\frown}{XY}=\frac{2}{3}\pi\), and \(r = WX\).

Step2: Substitute the values into the arc - length formula

Substitute \(n = 20\), \(s=\frac{2}{3}\pi\) into \(s=\frac{n\pi r}{180}\). We get \(\frac{2}{3}\pi=\frac{20\pi r}{180}\).

Step3: Solve for \(r\)

First, simplify the right - hand side of the equation: \(\frac{20\pi r}{180}=\frac{\pi r}{9}\).
So, \(\frac{2}{3}\pi=\frac{\pi r}{9}\).
Multiply both sides of the equation by \(9\) to get \(6\pi=\pi r\).
Divide both sides by \(\pi\), then \(r = 6\).

Answer:

\(6\)