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

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

Answer: (B)

Verified step by step guidance

1. Recall that the radian measure of a central angle in a circle is given by the ratio of the length of the arc it subtends to the radius of the circle, but more simply, the full circle corresponds to an angle of \$2\pi$ radians.
2. Since the entire circumference corresponds to \$2\pi\( radians, if arc CD is one-fourth of the circumference, then the central angle subtended by arc CD is one-fourth of \)2\pi$ radians.
3. Express this relationship mathematically as: \(\text{Central angle} = \frac{1}{4} \times 2\pi\).
4. Simplify the expression by multiplying the fraction and \$2\pi\(: \)\frac{1}{4} \times 2\pi = \frac{2\pi}{4}$.
5. Finally, reduce the fraction \(\frac{2\pi}{4}\) to its simplest form, which gives the radian measure of the central angle subtended by arc CD.