1. **Problem statement:** Find the missing side $x$ in the right triangle with legs $\sqrt{2}$ and $x$, and hypotenuse $\sqrt{10}$. 2. **Formula used:** In a right triangle, by the Pythagorean theorem, the sum of the squares of the legs equals the square of the hypotenuse: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are legs, and $c$ is the hypotenuse. 3. **Apply the formula:** Let $a = \sqrt{2}$, $b = x$, and $c = \sqrt{10}$. Then $$ (\sqrt{2})^2 + x^2 = (\sqrt{10})^2 $$ 4. **Simplify squares:** $$ 2 + x^2 = 10 $$ 5. **Isolate $x^2$:** $$ x^2 = 10 - 2 $$ $$ x^2 = 8 $$ 6. **Take the square root:** $$ x = \sqrt{8} $$ 7. **Simplify the radical:** $$ x = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} $$ **Final answer:** $$ x = 2\sqrt{2} $$