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 the arc - length formula

The formula for the length of an arc \(s = r\theta\) (where \(r\) is the radius 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 \(r = 6\ cm\).

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

Using the arc - length formula \(s_{MN}=r\theta\), substitute \(r = 6\) and \(\theta=\frac{5\pi}{12}\). Then \(s_{MN}=6\times\frac{5\pi}{12}=\frac{5\pi}{2}\approx\frac{5\times3.14}{2}=7.85\ cm\).

Step3: Calculate the length of arc \(LMN\)

Since \(s_{LMN}=s_{LM}+s_{MN}\), and \(s_{LM} = 6.3\ cm\), \(s_{MN}\approx7.85\ cm\). Then \(s_{LMN}=6.3 + 7.85=14.15\approx14.2\ cm\).

Answer:

14.2 cm