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 full angle in radians

A full circle has a central angle of \(2\pi\) radians (since the circumference corresponds to a full rotation, and the radian measure of a full angle is \(2\pi\)).

Step2: Calculate the central angle for the arc

The arc CD is \(\frac{2}{3}\) of the circumference. So the central angle \(\theta\) corresponding to arc CD is \(\frac{2}{3}\) of the full angle (\(2\pi\) radians).
Mathematically, \(\theta=\frac{2}{3}\times2\pi=\frac{4\pi}{3}\) radians.

Answer:

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