The area of triangle \(ABC\) is calculated using the formula for the area of a right triangle:

\(\text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2\sqrt{3} \times 2 = 2\sqrt{3}\)

The angle \(A\) can be found using the tangent function:

\(\tan A = \frac{2\sqrt{3}}{2} = \sqrt{3}\)

Thus, \(A = \frac{\pi}{3}\) radians (or 60 degrees).

The area of the sector \(ACD\) is given by:

\(\text{Area of sector } ACD = \frac{1}{2} \times r^2 \times \theta = \frac{1}{2} \times 2^2 \times \frac{\pi}{3} = \frac{2\pi}{3}\)

The area of the shaded region \(BDC\) is the area of triangle \(ABC\) minus the area of sector \(ACD\):

\(\text{Shaded area } BDC = 2\sqrt{3} - \frac{2\pi}{3}\)