In circle \( M, \mathrm{~m} \angle N M O=20^{\circ} \) and the length of \( \overparen{N O}=\frac{13}{9} \pi \). Find the length of \( \overline{M N} \).
Answer
Step 4: Use the Pythagorean theorem to solve for the length of MN, given that angle NMO is a right triangle with legs MN and MO. The length of MN is \(\sqrt{r^2 - MO^2}\). Compute the result.
Step 1 ：Step 1: Determine the fraction of the circumference of circle M represented by the arc NO. This is equal to the ratio of the measure of angle NMO to 360°, which is \(\frac{20}{360}\).
Step 2 ：Step 2: Calculate the circumference of circle M by dividing the length of arc NO by the fraction obtained in Step 1: \(C_M = \frac{\frac{13}{9}\pi}{\frac{20}{360}}\).
Step 3 ：Step 3: Find the radius of circle M using the circumference formula \(C_M = 2 \pi r\), where \(r\) is the radius of the circle: \(r =\frac{C_M}{2\pi}\).
Step 4 ：Step 4: Use the Pythagorean theorem to solve for the length of MN, given that angle NMO is a right triangle with legs MN and MO. The length of MN is \(\sqrt{r^2 - MO^2}\). Compute the result.
