Question

in circle o, m∠poq = 160° and the area of the shaded sector = 16/9 π. find the length of op.

Explanation:

Step1: Recall the sector - area formula

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central angle of the sector and \(r\) is the radius of the circle. Here, \(\theta = 160^{\circ}\) and \(A=\frac{16}{9}\pi\), and \(r = OP\).
Substitute the values into the formula: \(\frac{16}{9}\pi=\frac{160^{\circ}}{360^{\circ}}\times\pi\times(OP)^{2}\).

Step2: Simplify the equation

First, simplify \(\frac{160^{\circ}}{360^{\circ}}=\frac{4}{9}\).
The equation becomes \(\frac{16}{9}\pi=\frac{4}{9}\pi\times(OP)^{2}\).
Divide both sides of the equation by \(\frac{4}{9}\pi\).
\(\frac{\frac{16}{9}\pi}{\frac{4}{9}\pi}=(OP)^{2}\).
Since \(\frac{\frac{16}{9}\pi}{\frac{4}{9}\pi}=\frac{16}{9}\times\frac{9}{4}=4\), we have \((OP)^{2}=4\).

Step3: Solve for \(OP\)

Take the square root of both sides. Since \(OP\) represents the length of a radius (a non - negative quantity), \(OP=\sqrt{4}\).

Answer:

\(2\)