1. **State the problem:** We need to find the length of leg $a$ in a right triangle where the hypotenuse is 12 and the other leg is 6. 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. 3. **Apply the formula:** Here, $b = 6$ and $c = 12$, so: $$a^2 + 6^2 = 12^2$$ 4. **Calculate squares:** $$a^2 + 36 = 144$$ 5. **Isolate $a^2$:** $$a^2 = 144 - 36$$ 6. **Simplify:** $$a^2 = 108$$ 7. **Find $a$ by taking the square root:** $$a = \sqrt{108}$$ 8. **Simplify the square root:** $$a = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}$$ **Final answer:** $$a = 6\sqrt{3}$$