1. **State the problem:** We have a right triangle with legs of lengths $9$ and $x$, and hypotenuse length $24$. We need to find the exact value of $x$. 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. **Apply the formula:** Here, $a=9$, $b=x$, and $c=24$. So, $$9^2 + x^2 = 24^2$$ 4. **Calculate squares:** $$81 + x^2 = 576$$ 5. **Isolate $x^2$:** $$x^2 = 576 - 81$$ $$x^2 = 495$$ 6. **Simplify $x^2$ if possible:** $$495 = 9 \times 55$$ So, $$x^2 = 9 \times 55$$ 7. **Take the square root:** $$x = \sqrt{9 \times 55} = \sqrt{9} \times \sqrt{55} = 3\sqrt{55}$$ 8. **Check if the side lengths form a Pythagorean triple:** A Pythagorean triple consists of three positive integers satisfying $a^2 + b^2 = c^2$. Here, $x = 3\sqrt{55}$ is not an integer, so the side lengths do not form a Pythagorean triple. **Final answer:** $$x = 3\sqrt{55}$$ **Do the side lengths form a Pythagorean triple?** No.