Question

in circle x, m∠yxz = 80° and the area of the shaded sector = 8π. find the length of (overline{xy}).

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(\theta\) is the central angle and \(r\) is the radius of the circle. Here, \(\theta = 80^{\circ}\) and \(A = 8\pi\).
Substituting the values into the formula: \(8\pi=\frac{80}{360}\times\pi r^{2}\).

Step2: Simplify the equation

First, cancel out \(\pi\) from both sides of the equation \(8\pi=\frac{80}{360}\times\pi r^{2}\). We get \(8=\frac{80}{360}r^{2}\).
Then, simplify \(\frac{80}{360}=\frac{2}{9}\). So the equation becomes \(8=\frac{2}{9}r^{2}\).

Step3: Solve for \(r^{2}\)

Multiply both sides of the equation \(8=\frac{2}{9}r^{2}\) by \(\frac{9}{2}\).
\(r^{2}=8\times\frac{9}{2}\).
\(r^{2}=36\).

Step4: Solve for \(r\)

Take the square - root of both sides. Since \(r>0\) (radius of a circle), \(r = 6\).

Answer:

\(XY = 6\)