Question

in circle v, \\(\overarc{wyx}\\) is highlighted. \\(\angle wvx\\) measures \\(96^\circ\\).

what fraction of the circle is highlighted?
simplify your answer.

which expression represents the length of \\(\overarc{wyx}\\) in inches?
\\(\frac{2}{5}(2\pi)\\) \\(\frac{11}{15}(4\pi)\\) \\(\frac{2}{3}(4\pi)\\) \\(\frac{11}{15}(2\pi)\\)

Explanation:

Step1: Find unhighlighted arc angle

The unhighlighted arc $\widehat{WX}$ corresponds to central angle $\angle WVX = 96^\circ$.

Step2: Calculate highlighted arc angle

A full circle is $360^\circ$. Subtract the unhighlighted angle:
$360^\circ - 96^\circ = 264^\circ$

Step3: Find fraction of highlighted circle

Divide highlighted angle by full circle angle, then simplify:
$\frac{264}{360} = \frac{264 \div 24}{360 \div 24} = \frac{11}{15}$

Step4: Arc length formula setup

Arc length formula is $\frac{\text{arc angle}}{360^\circ} \times 2\pi r$, where $r=2$ in.

Step5: Substitute values for $\widehat{WYX}$

$\frac{264}{360} \times 2\pi(2) = \frac{11}{15} \times 4\pi = \frac{11}{15}(4\pi)$

Answer:

Fraction of the circle highlighted: $\frac{11}{15}$
Expression for length of $\widehat{WYX}$: $\frac{11}{15}(4\pi)$