1. **Stating the problem:** We are given a right triangle with sides 20 and 25, and a side $x$ opposite the right angle. We want to find the value of $x$. 2. **Formula used:** For a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the other two sides. 3. **Identify the sides:** Here, the sides given are 20 and 25, and $x$ is opposite the right angle, so $x$ is the hypotenuse. 4. **Apply the Pythagorean theorem:** $$20^2 + 25^2 = x^2$$ 5. **Calculate the squares:** $$400 + 625 = x^2$$ 6. **Sum the squares:** $$1025 = x^2$$ 7. **Solve for $x$ by taking the square root:** $$x = \sqrt{1025}$$ 8. **Simplify the square root if possible:** Since 1025 = 25 \times 41, $$x = \sqrt{25 \times 41} = \sqrt{25} \times \sqrt{41} = 5\sqrt{41}$$ **Final answer:** $$x = 5\sqrt{41}$$ This is the length of the side opposite the right angle in the triangle.