To determine the radian measure of the central angle corresponding to an arc that is [tex]\(\frac{1}{4}\)[/tex] of the circumference of a circle, follow these steps:
1. Understand the Relationship Between the Arc and the Circumference: Since arc CD is [tex]\(\frac{1}{4}\)[/tex] of the circumference, this means that the central angle subtended by arc CD would also be [tex]\(\frac{1}{4}\)[/tex] of the total central angle in a circle.
2. Recall the Full Circumference Central Angle: In a circle, the total central angle corresponding to the complete circumference is [tex]\(2\pi\)[/tex] radians.
3. Calculate the Central Angle for Arc CD:
[tex]\[
\text{Central angle for arc CD} = \frac{1}{4} \times 2\pi \text{ radians}
\][/tex]
4. Simplify the Expression:
[tex]\[
\frac{1}{4} \times 2\pi = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians}
\][/tex]
Thus, the radian measure of the central angle corresponding to arc CD is [tex]\(\frac{\pi}{2}\)[/tex] radians.
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{\pi}{2} \text{ radians}}
\][/tex]
