Let's solve the question step by step:
1. Understanding the Problem:
- We are given that Arc CD is [tex]\(\frac{1}{4}\)[/tex] of the circumference of a circle.
- We need to find the radian measure of the central angle corresponding to Arc CD.
2. Full Circle and Circumference:
- Recall that the circumference, [tex]\(C\)[/tex], of a circle is given by the formula [tex]\(C = 2\pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the circle.
- A full circle encompasses [tex]\(2\pi\)[/tex] radians.
3. Proportion of the Circle:
- Since Arc CD is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the angle subtended by Arc CD at the center of the circle will also be [tex]\(\frac{1}{4}\)[/tex] of the full central angle.
4. Calculating the Central Angle:
- The central angle for the full circle is [tex]\(2\pi\)[/tex] radians.
- Therefore, the central angle for Arc CD is [tex]\(\frac{1}{4} \times 2\pi\)[/tex].
5. Simplifying the Result:
- Calculate [tex]\(\frac{1}{4} \times 2\pi = \frac{2\pi}{4} = \frac{\pi}{2}\)[/tex].
Thus, the radian measure of the central angle corresponding to Arc CD is [tex]\(\frac{\pi}{2}\)[/tex] radians.
From the given choices:
[tex]\[
\frac{\pi}{4} \text{ radians,} \quad \frac{\pi}{2} \text{ radians,} \quad 2\pi \text{ radians,} \quad 4\pi \text{ radians}
\][/tex]
The correct answer is:
[tex]\[
\frac{\pi}{2} \text{ radians.}
\][/tex]
