Question

in the figure shown, an arc is subtended by a central angle measuring \frac{2\pi}{3} radians. what fraction of the circumference is this arc?

Explanation:

Step1: Recall the formula for arc length

The formula for arc length \(s = r\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians), and the circumference \(C = 2\pi r\).

Step2: Find the fraction

We want to find \(\frac{s}{C}\). Substitute \(s=r\theta\) and \(C = 2\pi r\) into the fraction. \(\frac{s}{C}=\frac{r\theta}{2\pi r}\). Cancel out the \(r\) terms. Given \(\theta=\frac{2\pi}{3}\), then \(\frac{s}{C}=\frac{\frac{2\pi}{3}}{2\pi}\).

Step3: Simplify the fraction

\(\frac{\frac{2\pi}{3}}{2\pi}=\frac{2\pi}{3}\times\frac{1}{2\pi}\). The \(2\pi\) terms cancel out, and we get \(\frac{1}{3}\).

Answer:

\(\frac{1}{3}\)