If arc AB is one-fourth of the circumference of a circle, what is the radian measure of the central angle subtended by arc AB?

• A. $\frac{1}{2}\pi$
• B. $\pi$
• C. $\frac{1}{4}\pi$
• D. $\frac{3}{4}\pi$

Answer: (C)

Verified step by step guidance

1. Recall that the circumference of a circle is given by the formula \(C = 2 \\pi r\), where \(r\) is the radius of the circle.
2. Understand that the central angle in radians corresponding to an arc length \(s\) on a circle of radius \(r\) is given by \(\theta = \frac{s}{r}\).
3. Since arc AB is one-fourth of the circumference, express the arc length as \(s = \frac{1}{4} \times 2 \\pi r = \frac{1}{2} \\pi r\).
4. Substitute the arc length \(s = \frac{1}{2} \\pi r\) into the central angle formula \(\theta = \frac{s}{r}\) to get \(\theta = \frac{\frac{1}{2} \\pi r}{r}\).
5. Simplify the expression by canceling \(r\) to find the radian measure of the central angle: \(\theta = \frac{1}{2} \\pi\) radians.