Question

the circle below has center c, and its radius is 3 cm. given that m∠dce = 100°, find the length of the major arc dfe. give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc dfe :

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 central angle from degrees to radians. The measure of the central angle of the minor arc $\overset{\frown}{DE}$ is $100^{\circ}$. The measure of the central angle of the major arc $\overset{\frown}{DFE}$ is $360^{\circ}-100^{\circ}=260^{\circ}$. To convert $260^{\circ}$ to radians, use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So, $\theta = 260\times\frac{\pi}{180}=\frac{13\pi}{9}$ radians.

Step2: Calculate the arc - length

Given that the radius $r = 3$ cm. Using the arc - length formula $s=r\theta$, substitute $r = 3$ and $\theta=\frac{13\pi}{9}$. Then $s=3\times\frac{13\pi}{9}=\frac{13\pi}{3}$ cm.

Answer:

$\frac{13\pi}{3}$ cm