(a) To find angle BAD, use the formula for the area of a sector: \(\frac{1}{2} \times r^2 \times \theta = 10\). Here, \(r = 4\) cm and \(\theta\) is the angle in radians.

\(\frac{1}{2} \times 4^2 \times \theta = 10\)

\(8 \theta = 10\)

\(\theta = \frac{10}{8} = 1.25\) radians

(b) To find the perimeter of the shaded region, calculate the arc length BD and the lengths BC and CD.

Arc BD = \(4 \times 1.25 = 5\) cm

Using \(BC = 4 \tan(1.25)\), calculate \(BC\).

\(BC \approx 12.04\) cm

Using \(CD = \frac{4}{\cos(1.25)} - 4\), calculate \(CD\).

\(CD \approx 8.69\) cm

Perimeter = Arc BD + BC + CD = \(5 + 12.04 + 8.69 = 25.7\) cm