To determine the radian measure of the central angle for an arc CD, which is [tex]\(\frac{1}{4}\)[/tex] of the circumference of a circle, let's follow these steps:
1. Understand the relationship between arc length and the central angle:
- The circumference of the entire circle is [tex]\(2\pi\)[/tex] radians in terms of the central angle.
- Given that arc CD is [tex]\(\frac{1}{4}\)[/tex] of the total circumference, the central angle corresponding to arc CD will be [tex]\(\frac{1}{4}\)[/tex] of the full circle's central angle.
2. Calculate the central angle in radians:
- Since the full circle in radians is [tex]\(2\pi\)[/tex],
- The central angle corresponding to arc CD is [tex]\(\frac{1}{4} \times 2\pi\)[/tex].
3. Perform the fraction multiplication:
- [tex]\(\frac{1}{4} \times 2\pi = \frac{2\pi}{4} = \frac{\pi}{2}\)[/tex].
So, the radian measure of the central angle corresponding to arc CD, which is [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle, is [tex]\(\frac{\pi}{2}\)[/tex] radians.
Hence, the correct answer is [tex]\(\frac{\pi}{2}\)[/tex] radians.
