Question

in circle t, tu = 2 and the area of shaded sector = \frac{7}{9}\pi. find m\angle utv.

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 \(A\) is the area of the sector, \(\theta\) is the central angle (in degrees), and \(r\) is the radius of the circle.

Step2: Substitute the given values into the formula

We are given that \(r = TU=2\) and \(A=\frac{7}{9}\pi\). Substituting these into the formula \(A=\frac{\theta}{360}\times\pi r^{2}\), we get \(\frac{7}{9}\pi=\frac{\theta}{360}\times\pi\times(2)^{2}\).
First, simplify the right - hand side: \(\frac{\theta}{360}\times\pi\times4=\frac{\pi\theta}{90}\).
So, our equation becomes \(\frac{7}{9}\pi=\frac{\pi\theta}{90}\).
Since \(\pi
eq0\), we can divide both sides of the equation by \(\pi\): \(\frac{7}{9}=\frac{\theta}{90}\).

Step3: Solve for \(\theta\)

Cross - multiply: \(9\theta=7\times90\).
Then \(9\theta = 630\).
Divide both sides by 9: \(\theta=\frac{630}{9}=70\).

Answer:

\(m\angle UTV = 70^{\circ}\)