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…

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle with radius $r$ and central - angle $\theta$ (in degrees) is $s=\frac{\theta}{360}\times2\pi r$.

Step2: First, find the length of arc $MN$

Given $r = 6$ cm and $\theta=75^{\circ}$, then the length of arc $MN$ is $s_{MN}=\frac{75}{360}\times2\pi\times6$. [ \begin{align*} s_{MN}&=\frac{75}{360}\times12\pi\ &=\frac{75\times12\pi}{360}\ &=\frac{75\pi}{30}\ & = 2.5\pi\approx2.5\times3.14 = 7.85\text{ cm} \end{align*} ]

Step3: Calculate the length of arc $LMN$

The length of arc $LM$ is given as $s_{LM}=6.3$ cm. The length of arc $LMN$ is $s_{LMN}=s_{LM}+s_{MN}$. [ \begin{align*} s_{LMN}&=6.3 + 7.85\ &=14.15\text{ cm}\approx14.2\text{ cm} \end{align*} ]

Answer:

14.2 cm