Question

arc cd is \\(\frac{2}{3}\\) of the circumference of a circle. what is the radian measure of the central angle?\\(\frac{2\pi}{3}\\) radians\\(\frac{3\pi}{4}\\) radians\\(\frac{4\pi}{3}\\) radians\\(\frac{3\pi}{2}\\) radians

Explanation:

Step1: Recall the total radians in a circle

A full circle has a central angle of \(2\pi\) radians.

Step2: Calculate the central angle for arc CD

Since arc CD is \(\frac{2}{3}\) of the circumference, the central angle (in radians) is \(\frac{2}{3}\times2\pi\).
Simplify the expression: \(\frac{2}{3}\times2\pi=\frac{4\pi}{3}\) radians.

Answer:

\(\frac{4\pi}{3}\) radians (corresponding to the option " \(\boldsymbol{\frac{4\pi}{3}}\) radians")