1. **State the problem:** We have a right triangle with legs of lengths 2 cm and $x$ cm, and a hypotenuse of 10 cm. We need to find the perimeter of the triangle. 2. **Use the Pythagorean theorem:** For a right triangle, the relationship between the legs and the hypotenuse is given by: $$x^2 + 2^2 = 10^2$$ 3. **Calculate $x$:** $$x^2 + 4 = 100$$ $$x^2 = 100 - 4$$ $$x^2 = 96$$ $$x = \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$$ 4. **Find the perimeter:** The perimeter $P$ is the sum of all sides: $$P = 2 + x + 10$$ Substitute $x$: $$P = 2 + 4\sqrt{6} + 10$$ $$P = 12 + 4\sqrt{6}$$ 5. **Final answer:** The perimeter of the triangle is $$\boxed{12 + 4\sqrt{6} \text{ cm}}$$