Question

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

Step2: Calculate the central angle for the arc

The arc CD is \(\frac{1}{4}\) of the circumference. So, the central angle (in radians) is \(\frac{1}{4}\) of \(2\pi\).
\[
\frac{1}{4} \times 2\pi=\frac{2\pi}{4}=\frac{\pi}{2}
\]

Answer:

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