Let the length of the hypotenuse of the right-angled triangle be \(x\) cm. ∴ The lengths of the other two sides are \((x - 6)\) cm and \((x - 12)\) cm. In a right-angled triangle, the sum of the squares of the two sides adjacent to the right angle equals the square of the hypotenuse. So, according to the question: \[ (x - 6)^2 + (x - 12)^2 = x^2 \] \[ x^2 - 12x + 36 + x^2 - 24x + 144 = x^2 \Rightarrow 2x^2 - 36x + 180 - x^2 = 0 \Rightarrow x^2 - 36x + 180 = 0 \] \[ x^2 - (30 + 6)x + 180 = 0 \Rightarrow x^2 - 30x - 6x + 180 = 0 \Rightarrow x(x - 30) - 6(x - 30) = 0 \Rightarrow (x - 30)(x - 6) = 0 \] ∴ Either \(x = 30\) or \(x = 6\) But if the hypotenuse is 6 cm, the triangle cannot exist. ∴ The hypotenuse is 30 cm. ∴ The other two sides are: \(30 - 6 = 24\) cm and \(30 - 12 = 18\) cm ∴ Area of the triangle \[ = \frac{1}{2} \times 24 \times 18 = 216 \text{ cm} \] (Answer)