Question

in circle o, the length of radius ol is 6 cm and the length of arc lm is 6.3 cm. the measure of angle mon is 75°. rounded to the nearest tenth of a centimeter, what is the length of arc lmn? 7.9 cm 10.2 cm 12.6 cm 14.2 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 $\angle MON = 75^{\circ}$ to radians. We know that to convert degrees to radians, we use the formula $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$. So, $\theta=\frac{75\pi}{180}=\frac{5\pi}{12}$ radians and $r = 6$ cm.

Step2: Calculate the length of arc $MN$

Using the arc - length formula $s = r\theta$, with $r = 6$ cm and $\theta=\frac{5\pi}{12}$, we have $s_{MN}=6\times\frac{5\pi}{12}=\frac{5\pi}{2}\approx\frac{5\times3.14}{2}=7.85$ cm.

Step3: Calculate the length of arc $LMN$

We know that the length of arc $LM = 6.3$ cm. The length of arc $LMN$ is the sum of the length of arc $LM$ and the length of arc $MN$. So, $s_{LMN}=6.3 + 7.85=14.15\approx14.2$ cm.

Answer:

14.2 cm