Question

in circle t, m∠utv = 72° and the length of (overset{\frown}{uv}=\frac{6}{5}pi). find the length of (overline{tu}).

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{n}{360}\times2\pi r\), where \(n\) is the central angle and \(r\) is the radius of the circle. Here, \(n = m\angle UTV=72^{\circ}\), \(s=\overset{\frown}{UV}=\frac{6}{5}\pi\), and \(r = TU\).
Substitute the values into the formula: \(\frac{6}{5}\pi=\frac{72}{360}\times2\pi r\).

Step2: Simplify the equation

First, simplify \(\frac{72}{360}=\frac{1}{5}\).
The equation becomes \(\frac{6}{5}\pi=\frac{1}{5}\times2\pi r\).
Multiply both sides of the equation by \(5\) to get \(6\pi = 2\pi r\).

Step3: Solve for \(r\)

Divide both sides of the equation \(6\pi = 2\pi r\) by \(2\pi\).
\(r=\frac{6\pi}{2\pi}\).

Answer:

\(3\)