To determine the radian measure of a central angle corresponding to the arc \(\overline{CD}\), given that the arc is \(\frac{1}{4}\) of the circumference of a circle, follow these steps:
1. Understand the relationship between arc length and central angle:
The radian measure of a central angle is directly proportional to the length of the arc it subtends. Specifically, if the arc length is a fraction of the entire circumference, the central angle will be the same fraction of \(2\pi\) radians (which represents the angle for the full circle).
2. Given information:
Arc \(\overline{CD}\) is \(\frac{1}{4}\) of the circumference of the circle.
3. Determine the fraction of the full circle:
Since the arc \(\overline{CD}\) is \(\frac{1}{4}\) of the circumference, the central angle that subtends this arc is \(\frac{1}{4}\) of the full circle's angle.
4. Full circle in radians:
The angle for a full circle is \(2\pi\) radians.
5. Calculate the central angle in radians:
Since the arc is \(\frac{1}{4}\) of the circumference, the central angle is:
[tex]\[
\frac{1}{4} \times 2\pi = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians}
\][/tex]
Therefore, the radian measure of the central angle for arc \(\overline{CD}\) is \(\frac{\pi}{2}\) radians.
The correct answer is [tex]\[
\boxed{\frac{\pi}{2} \text{ radians}}
\][/tex]
