Question

in the diagram below, what is the approximate length of the minor arc $widehat{xy}$? 27 cm 60° a. 14.1 cm b. 56.5 cm c. 25.7 cm d. 28.3 cm

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc of a circle is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. First, convert the angle from degrees to radians. We know that to convert degrees to radians, we use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 60^{\circ}$, then $\theta_{rad}=60\times\frac{\pi}{180}=\frac{\pi}{3}$ radians, and $r = 27$ cm.

Step2: Calculate the arc - length

Substitute $r = 27$ cm and $\theta=\frac{\pi}{3}$ into the arc - length formula $s=r\theta$. So $s = 27\times\frac{\pi}{3}$.
\[

$$\begin{align*} s&=9\pi\\ s&\approx9\times3.14\\ s&\approx28.26\approx28.3\text{ cm} \end{align*}$$

\]

Answer:

D. 28.3 cm