(i) To find the perimeter of the shaded region PQT, we need to calculate the lengths of PT, QT, and the arc PQ.

The length of PT is given by the tangent formula: \(PT = r \tan \alpha\).

The length of QT is \(QT = OT - OQ = \frac{r}{\cos \alpha} - r\).

The length of the arc PQ is \(r \alpha\).

Therefore, the perimeter of the shaded region PQT is \(r \tan \alpha + \frac{r}{\cos \alpha} - r + r \alpha\).

(ii) For \(\alpha = \frac{1}{3} \pi\) and \(r = 10\):

The area of triangle POT is \(\frac{1}{2} \times 10 \times 10 \times \tan \frac{\pi}{3} = 50 \sqrt{3}\).

The area of sector POQ is \(\frac{1}{2} \times 10^2 \times \frac{1}{3} \pi = \frac{50 \pi}{3}\).

The area of the shaded region is \(50 \sqrt{3} - \frac{50 \pi}{3} \approx 34\) (to 2 significant figures).