1. **Problem:** Find the length of segment NM given that segments KL and MN are parts of chord KM in a circle, with lengths $x - 3$ and $x + 3$ respectively, and the entire chord KM has length 16. 2. **Formula and rule:** The sum of parts of a chord equals the whole chord length. So, $$KL + MN = KM$$ 3. **Set up the equation:** $$ (x - 3) + (x + 3) = 16 $$ 4. **Simplify the equation:** $$ x - 3 + x + 3 = 16 $$ $$ 2x = 16 $$ 5. **Solve for $x$:** $$ x = \frac{16}{2} = 8 $$ 6. **Find NM:** $$ NM = x + 3 = 8 + 3 = 11 $$ **Answer:** The length of segment NM is 11.