1. **State the problem:** Find the value of $x$ in the right triangle where the sides satisfy the Pythagorean theorem. 2. **Formula:** The Pythagorean theorem states that for a right triangle with legs $a$ and $b$, and hypotenuse $c$, the relationship is: $$a^2 + b^2 = c^2$$ 3. **Apply the formula to the first example:** Given: $a=3$, $b=x$, $c=9$ $$3^2 + x^2 = 9^2$$ Calculate squares: $$9 + x^2 = 81$$ Isolate $x^2$: $$x^2 = 81 - 9$$ $$x^2 = 72$$ Take the square root: $$x = \sqrt{72}$$ Simplify the radical: $$x = \sqrt{36 \times 2} = 6\sqrt{2}$$ 4. **Explanation:** We used the Pythagorean theorem to find the missing side $x$. We squared the known sides, subtracted to isolate $x^2$, then took the square root and simplified. **Final answer:** $$x = 6\sqrt{2}$$