To find the radian measure of the central angle corresponding to an arc that is [tex]\(\frac{1}{4}\)[/tex] of the circumference of a circle, we need to understand the relationship between the arc length, the radius of the circle, and the central angle.
1. Understanding the circumference of a circle:
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.
2. Understanding the arc length and the fraction of the circumference:
If arc CD is [tex]\(\frac{1}{4}\)[/tex] of the circumference, then the length of arc CD is:
[tex]\[
\text{Length of arc CD} = \frac{1}{4} \times C = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}
\][/tex]
3. The relationship between the arc length and the central angle in radians:
The length of an arc (s) is related to the central angle ([tex]\(\theta\)[/tex]) in radians and the radius of the circle (r) by the formula:
[tex]\[
s = r\theta
\][/tex]
Substituting the length of arc CD into this formula gives:
[tex]\[
\frac{\pi r}{2} = r\theta
\][/tex]
4. Solving for the central angle [tex]\(\theta\)[/tex]:
To find [tex]\(\theta\)[/tex], we can solve the equation:
[tex]\[
\frac{\pi r}{2} = r\theta
\][/tex]
Dividing both sides by [tex]\(r\)[/tex] (assuming [tex]\(r \neq 0\)[/tex]):
[tex]\[
\frac{\pi}{2} = \theta
\][/tex]
Therefore, the radian measure of the central angle corresponding to an arc that is [tex]\(\frac{1}{4}\)[/tex] of the circumference of a circle is:
[tex]\[
\boxed{\frac{\pi}{2} \text{ radians}}
\][/tex]
