Question

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

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. Also, the circumference of a circle is $C = 2\pi r$. Given that the arc length $s=\frac{2}{3}C$.

Step2: Substitute $C = 2\pi r$ into the arc - length relationship

Since $s=\frac{2}{3}C$ and $C = 2\pi r$, then $s=\frac{2}{3}\times2\pi r=\frac{4\pi r}{3}$. And since $s = r\theta$, we can set $\frac{4\pi r}{3}=r\theta$.

Step3: Solve for $\theta$

Divide both sides of the equation $\frac{4\pi r}{3}=r\theta$ by $r$ (assuming $r
eq0$). We get $\theta=\frac{4\pi}{3}$ radians.

Answer:

$\frac{4\pi}{3}$ radians