In circle \( L, \mathrm{~m} \angle M L N=20^{\circ} \) and the length of \( M N=\frac{7}{9} \pi \). Find the length of \( \overline{L M} \).

Step 1 ：Step 1: Find the circumference of circle L using the given arc MN: \( C = \frac{7}{9} \pi \times 9 = 7\pi \)

Step 2 ：Step 2: Determine the radius of circle L: \( r = \frac{C}{2\pi} = \frac{7\pi}{2\pi} = \frac{7}{2} \)

Step 3 ：Step 3: Calculate the length of LM which is equal to the radius: \( LM = r = \frac{7}{2} \)