1. **State the problem:** We have a right triangle with legs measuring 10 cm and 24 cm, and we need to find the hypotenuse $x$ using Pythagoras' theorem. 2. **Formula:** Pythagoras' theorem states that in a right triangle, the square of the hypotenuse ($x$) is equal to the sum of the squares of the other two sides: $$x^2 = a^2 + b^2$$ where $a$ and $b$ are the legs of the triangle. 3. **Apply the formula:** Substitute $a = 10$ cm and $b = 24$ cm: $$x^2 = 10^2 + 24^2$$ 4. **Calculate squares:** $$x^2 = 100 + 576$$ 5. **Sum the squares:** $$x^2 = 676$$ 6. **Find $x$ by taking the square root:** $$x = \sqrt{676}$$ 7. **Simplify the square root:** $$x = 26$$ **Final answer:** The length of the hypotenuse $x$ is 26 cm.