Question

in the circle shown, an arc is subtended by a central angle measuring \frac{5\pi}{6} radians. what fraction of the circumference is this arc?

Explanation:

Step1: Recall the formula for the circumference of a circle

The circumference of a circle is \(C = 2\pi r\). The length of an arc \(s\) subtended by a central angle \(\theta\) (in radians) is given by \(s=r\theta\).

Step2: Find the fraction of the arc length to the circumference

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:

$$ \frac{s}{C}=\frac{\theta}{2\pi} $$

Given \(\theta=\frac{5\pi}{6}\), substitute \(\theta\) into the formula:

$$ \frac{s}{C}=\frac{\frac{5\pi}{6}}{2\pi}=\frac{5\pi}{6}\times\frac{1}{2\pi} $$

Cancel out the \(\pi\) terms:

$$ \frac{s}{C}=\frac{5}{12} $$

Answer:

\(\frac{5}{12}\)