Question

in circle t, m∠utv = 80° and the length of ⌢uv = 4/3π. find the length of tu.

Explanation:

Step1: Use arc - length formula

The arc - length formula is \(l=\frac{n\pi r}{180}\), where \(l\) is the arc - length, \(n\) is the central angle in degrees, and \(r\) is the radius of the circle. Here, \(l = \frac{4}{3}\pi\), \(n = 80^{\circ}\), and \(r=TU\).
Substitute the values into the formula: \(\frac{4}{3}\pi=\frac{80\pi\times TU}{180}\).

Step2: Solve for \(TU\)

First, simplify the right - hand side of the equation \(\frac{80\pi\times TU}{180}=\frac{4\pi\times TU}{9}\).
So, our equation becomes \(\frac{4}{3}\pi=\frac{4\pi\times TU}{9}\).
Multiply both sides of the equation by \(\frac{9}{4\pi}\) (to isolate \(TU\)).
\(TU=\frac{4}{3}\pi\times\frac{9}{4\pi}\).
Cancel out the common factors: \(\pi\) cancels, and 4 cancels. We get \(TU = 3\).

Answer:

\(3\)