1. **Stating the problem:** We are given a right triangle with one leg of length 6 and the hypotenuse of length $3\sqrt{13}$. We want to find the length of the other leg. 2. **Formula used:** In a right triangle, the Pythagorean theorem states: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Assign known values:** Let the unknown leg be $x$. Then: $$6^2 + x^2 = (3\sqrt{13})^2$$ 4. **Calculate squares:** $$36 + x^2 = 9 \times 13$$ $$36 + x^2 = 117$$ 5. **Isolate $x^2$:** $$x^2 = 117 - 36$$ $$x^2 = 81$$ 6. **Solve for $x$:** $$x = \sqrt{81}$$ $$x = 9$$ **Final answer:** The length of the other leg is $9$.