Question

in circle v, ( mangle wvx = 45^{circ} ) and the area of shaded sector ( =\frac{1}{2}pi ). find the length of ( overparen{wx} ). express your answer as a fraction times ( pi ).

Explanation:

Step1: Use the sector - area formula

The formula for the area of a sector is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central angle and \(r\) is the radius of the circle. Given \(\theta = 45^{\circ}\) and \(A=\frac{1}{2}\pi\).
Substitute into the formula: \(\frac{1}{2}\pi=\frac{45^{\circ}}{360^{\circ}}\times\pi r^{2}\).
Simplify \(\frac{45^{\circ}}{360^{\circ}}=\frac{1}{8}\), so \(\frac{1}{2}\pi=\frac{1}{8}\pi r^{2}\).
Divide both sides by \(\pi\): \(\frac{1}{2}=\frac{1}{8}r^{2}\).
Multiply both sides by \(8\): \(r^{2}=4\), then \(r = 2\) (since \(r>0\)).

Step2: Use the arc - length formula

The formula for the arc length \(l\) is \(l=\frac{\theta}{360^{\circ}}\times2\pi r\).
Substitute \(\theta = 45^{\circ}\) and \(r = 2\) into the formula.
\(l=\frac{45^{\circ}}{360^{\circ}}\times2\pi\times2\).
Simplify \(\frac{45^{\circ}}{360^{\circ}}=\frac{1}{8}\), then \(l=\frac{1}{8}\times4\pi\).
\(l=\frac{1}{2}\pi\).

Answer:

The length of \(\overset{\frown}{WX}\) is \(\frac{1}{2}\pi\).