1. **State the problem:** We have a right triangle with hypotenuse 36 and base 12, and a smaller right triangle inside it with a segment labeled $a$ opposite the hypotenuse side 36. 2. **Identify the goal:** Find the length of segment $a$. 3. **Recall the Pythagorean theorem:** For a right triangle with legs $x$ and $y$ and hypotenuse $c$, the relation is $$x^2 + y^2 = c^2$$ 4. **Apply the theorem to the main triangle:** Let the height be $h$. Then: $$12^2 + h^2 = 36^2$$ $$144 + h^2 = 1296$$ 5. **Solve for $h^2$:** $$h^2 = 1296 - 144 = 1152$$ 6. **Calculate $h$:** $$h = \sqrt{1152} = \sqrt{256 \times 4.5} = 16 \sqrt{4.5} = 16 \times \sqrt{\frac{9}{2}} = 16 \times \frac{3}{\sqrt{2}} = \frac{48}{\sqrt{2}}$$ 7. **Simplify $h$ by rationalizing the denominator:** $$h = \frac{48}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{48 \sqrt{2}}{2} = 24 \sqrt{2}$$ 8. **Interpret segment $a$:** Since $a$ is opposite the hypotenuse side 36 and inside the triangle, it corresponds to the height $h$. 9. **Final answer:** $$a = 24 \sqrt{2}$$