Question

in circle w, m∠xwy = 135° and the length of ⌢xy = 3π. 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, \(\theta = 135^{\circ}\), \(s = 3\pi\), and \(r=WX\).

Step2: Substitute the values into the arc - length formula

Substitute \(s = 3\pi\) and \(\theta=135^{\circ}\) into \(s=\frac{\theta}{360^{\circ}}\times2\pi r\). We get \(3\pi=\frac{135^{\circ}}{360^{\circ}}\times2\pi r\).
First, simplify \(\frac{135^{\circ}}{360^{\circ}}=\frac{3}{8}\). So the equation becomes \(3\pi=\frac{3}{8}\times2\pi r\).
Then simplify \(\frac{3}{8}\times2\pi r=\frac{3\pi r}{4}\). So \(3\pi=\frac{3\pi r}{4}\).

Step3: Solve for \(r\)

Multiply both sides of the equation \(3\pi=\frac{3\pi r}{4}\) by \(\frac{4}{3\pi}\).
\(r = 4\).

Answer:

\(4\)