Arc CD is [tex]\frac{1}{4}[/tex] of the circumference of a circle. What is the radian measure of the central angle?

A. [tex]\frac{\pi}{4}[/tex] radians
B. [tex]\frac{\pi}{2}[/tex] radians
C. [tex]2 \pi[/tex] radians
D. [tex]4 \pi[/tex] radians



Answer :

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]