Question

arc cd is \\(\frac{1}{4}\\) of the circumference of a circle. what is the radian measure of the central angle? \\(\frac{\pi}{4}\\) radians \\(\frac{\pi}{2}\\) radians \\(2\pi\\) radians \\(4\pi\\) radians

Explanation:

Step1: Recall the total radians in a circle

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

Step2: Calculate the central angle for arc CD

Arc CD is \( \frac{1}{4} \) of the circumference. So the central angle for arc CD is \( \frac{1}{4} \) of the total central angle of the circle.
The total central angle is \( 2\pi \) radians. So we calculate \( \frac{1}{4} \times 2\pi \).
Simplify \( \frac{1}{4} \times 2\pi=\frac{2\pi}{4}=\frac{\pi}{2} \) radians.

Answer:

\(\frac{\pi}{2}\) radians (corresponding to the option: \(\boldsymbol{\frac{\pi}{2}}\) radians)