First, calculate the angles in triangle ABC:

\(\angle BAC = \sin^{-1}(3/5) \approx 36.87^\circ\)

\(\angle ABC = \sin^{-1}(4/5) \approx 53.13^\circ\)

\(\angle ACB = \pi/2\) (right angle)

Calculate the area of triangle ABC:

\(\Delta ABC = \frac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2\)

Calculate the areas of the sectors:

\(\text{Sector AEF} = \frac{1}{2} \times 3^2 \times 0.6435\)

\(\text{Sector BEG} = \frac{1}{2} \times 2^2 \times 0.9273\)

\(\text{Sector CFG} = \frac{1}{2} \times 1^2 \times 1.5708\)

Sum of sectors = \(\frac{1}{2} [3^2 \times 0.6435 + 2^2 \times 0.9273 + 1^2 \times 1.5708]\)

Sum of sectors = 5.536 \, \text{cm}^2

Area of shaded region EFG:

\(\text{Area of EFG} = \Delta ABC - \text{Sum of sectors} = 6 - 5.536 = 0.464 \, \text{cm}^2\)