To find the radian measure of the central angle corresponding to arc CD, we need to understand the relationship between the arc length and the angle in a circle.
Here’s a step-by-step breakdown of how to achieve this:
1. Understand the Circumference of a Circle in Radians:
The total circumference of a circle in radians is [tex]\(2\pi\)[/tex] radians. This is due to the fact that a full circle (360°) is equal to [tex]\(2\pi\)[/tex] radians.
2. Determine the Fraction Corresponding to Arc CD:
Arc CD is given to be [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle.
3. Calculate the Central Angle:
Since the total circumference in radians is [tex]\(2\pi\)[/tex] and arc CD is [tex]\(\frac{1}{4}\)[/tex] of that circumference, we can find the measure of the central angle by multiplying the fraction of the circumference by the total radians:
[tex]\[
\text{Central Angle} = \frac{1}{4} \times 2\pi
\][/tex]
4. Perform the Multiplication:
[tex]\[
\text{Central Angle} = \frac{1}{4} \times 2\pi = \frac{2\pi}{4} = \frac{\pi}{2}
\][/tex]
Thus, the radian measure of the central angle corresponding to arc CD is [tex]\(\frac{\pi}{2}\)[/tex] radians.
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{\pi}{2} \text{ radians}}
\][/tex]
