1. **Problem statement:** Given a circle with center $L$, points $J$, $K$, $M$, and $P$ on the circle, segment $JK$ passes through $N$ and is perpendicular to segment $ML$ at $N$. Given $NL=9$ and $LK=15$, find $NK$. 2. **Relevant formulas and rules:** - Since $L$ is the center and $J$, $K$, $M$, $P$ lie on the circle, $LJ=LK=LM=LP$ (radii). - $LK=15$ means the radius of the circle is 15. - $N$ lies on $JK$ and $ML$ is perpendicular to $JK$ at $N$, so $N$ is the foot of the perpendicular from $L$ to chord $JK$. - The perpendicular from the center to a chord bisects the chord. 3. **Find $NK$:** - Since $N$ is the midpoint of chord $JK$, $NK = NJ$. - We know $NL=9$ and $LK=15$. - Triangle $LNK$ is right-angled at $N$ because $NL$ is perpendicular to $JK$. 4. **Use the Pythagorean theorem in triangle $LNK$: $$LK^2 = LN^2 + NK^2$$** 5. Substitute known values: $$15^2 = 9^2 + NK^2$$ 6. Calculate squares: $$225 = 81 + NK^2$$ 7. Solve for $NK^2$: $$NK^2 = 225 - 81 = 144$$ 8. Take the square root: $$NK = \sqrt{144} = 12$$ **Final answer:** $NK = 12$