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?

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s = r\theta\) (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 \(\theta (in\ radians)=\frac{\pi}{180}\times\theta (in\ degrees)\). So, \(\theta=\frac{75\pi}{180}=\frac{5\pi}{12}\) radians. Given \(r = 6\ cm\). Then the length of arc \(MN\) is \(s_{MN}=r\theta=6\times\frac{5\pi}{12}=\frac{5\pi}{2}\approx 7.9\ cm\)

Step2: Calculate the length of arc \(LMN\)

The length of arc \(LMN\) is the sum of the length of arc \(LM\) and the length of arc \(MN\). We know that \(s_{LM}=6.3\ cm\) and \(s_{MN}\approx7.9\ cm\). Then \(s_{LMN}=s_{LM}+s_{MN}\)

Answer:

\(14.2\ cm\)