To find the length x in the geometric problem with a shaded area of 1200 cm², the areas of a triangle and rectangle are considered. The triangle's area is initially set up as A_triangle = (1/2)(x-1)(x), but it is corrected to A_triangle = (1/2)x². The equation combining both areas is A_rectangle + A_triangle = 2400, leading to the equation (1/2)x² + x = 1200. Solving this quadratic equation yields x = 48 cm as the valid solution. The final answer is x = 48 cm.