arc cd is $\frac{2}{3}$ of the circumference of a circle. what is the radian measure of the central…

arc cd is $\frac{2}{3}$ of the circumference of a circle. what is the radian measure of the central angle?\n$\frac{2pi}{3}$ radians\n$\frac{3pi}{4}$ radians\n$\frac{4pi}{3}$ radians\n$\frac{3pi}{2}$ radians

arc cd is $\frac{2}{3}$ of the circumference of a circle. what is the radian measure of the central angle?\n$\frac{2pi}{3}$ radians\n$\frac{3pi}{4}$ radians\n$\frac{4pi}{3}$ radians\n$\frac{3pi}{2}$ radians

Answer

Explanation:

Step1: Recall the relationship between arc - length, circumference and central - angle

The measure of a full - circle in radians is (2\pi) radians. If an arc is a fraction (f) of the circumference of a circle, the radian measure of the central angle (\theta) corresponding to that arc is given by (\theta = f\times2\pi).

Step2: Identify the fraction of the arc

We are given that the arc (CD) is (\frac{2}{3}) of the circumference of the circle, so (f=\frac{2}{3}).

Step3: Calculate the radian measure of the central angle

Substitute (f = \frac{2}{3}) into the formula (\theta=f\times2\pi). Then (\theta=\frac{2}{3}\times2\pi=\frac{4\pi}{3}) radians.

Answer:

C. (\frac{4\pi}{3}) radians