Question

in circle w, m∠xwy = 20° and the length of \\(\overarc{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 \( s=\frac{\theta}{360^{\circ}}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, \( \theta = m\angle XWY = 20^{\circ} \), \( s=\frac{2}{3}\pi \), and \( r = WX \) (since \( WX \) is a radius of circle \( W \)).

Step2: Substitute the known values into the formula

Substitute \( s = \frac{2}{3}\pi \), \( \theta=20^{\circ} \) into the arc length formula:
\[
\frac{2}{3}\pi=\frac{20^{\circ}}{360^{\circ}}\times2\pi r
\]
Simplify the right - hand side: \( \frac{20^{\circ}}{360^{\circ}}\times2\pi r=\frac{1}{18}\times2\pi r=\frac{\pi r}{9} \)
So we have the equation:
\[
\frac{2}{3}\pi=\frac{\pi r}{9}
\]

Step3: Solve for \( r \) (which is \( WX \))

Multiply both sides of the equation \( \frac{2}{3}\pi=\frac{\pi r}{9} \) by 9 to get rid of the denominator on the right - hand side:
\( 9\times\frac{2}{3}\pi=\pi r \)
Simplify the left - hand side: \( 6\pi=\pi r \)
Divide both sides by \( \pi \): \( r = 6 \)

Answer:

The length of \( \overline{WX} \) is \( 6 \).