1. **State the problem:** We have a right triangle with vertices K, L, and J, and a perpendicular segment KM from K to LJ. Given lengths are $KM=6$ and $MJ=8$. We need to find the length $JL$. 2. **Understand the setup:** Since $KM$ is perpendicular to $LJ$, $KM$ is the height from $K$ to the hypotenuse $LJ$ of triangle $KLJ$. 3. **Use the right triangle properties:** The segment $MJ=8$ lies on $LJ$, and $KM=6$ is the height from $K$ to $LJ$. The segment $LJ$ is composed of $LM + MJ$. 4. **Apply the geometric mean theorem (altitude rule):** In a right triangle, the altitude to the hypotenuse satisfies: $$KM^2 = LM \times MJ$$ 5. **Calculate $LM$:** $$6^2 = LM \times 8$$ $$36 = 8 LM$$ $$LM = \frac{36}{8} = 4.5$$ 6. **Find $LJ$:** $$LJ = LM + MJ = 4.5 + 8 = 12.5$$ **Final answer:** $$JL = 12.5$$