(i) The area of the segment is given by \(\frac{1}{2} r^2 \theta - \frac{1}{2} r^2 \sin \theta\). With \(r = 10\), this becomes \(50\theta - 50\sin \theta\).

The area of the circle is \(\pi r^2 = 100\pi\). The area of the segment is \(\frac{1}{5}\) of the circle, so:

\(50\theta - 50\sin \theta = \frac{1}{5} \times 100\pi\)

\(50\theta - 50\sin \theta = 20\pi\)

\(\theta - \sin \theta = \frac{2}{5}\pi\)

Thus, \(\theta = \frac{2}{5}\pi + \sin \theta\).

(ii) Using the iterative formula \(\theta_{n+1} = \frac{2}{5}\pi + \sin \theta_n\) with an initial value of \(\theta_0 = 2.1\):

\(\theta_1 = \frac{2}{5}\pi + \sin 2.1 \approx 2.1198\)

\(\theta_2 = \frac{2}{5}\pi + \sin 2.1198 \approx 2.1097\)

\(\theta_3 = \frac{2}{5}\pi + \sin 2.1097 \approx 2.1149\)

\(\theta_4 = \frac{2}{5}\pi + \sin 2.1149 \approx 2.1122\)

\(\theta_5 = \frac{2}{5}\pi + \sin 2.1122 \approx 2.1111\)

Thus, \(\theta \approx 2.11\) radians.

To find the length of AB, use the formula:

\(AB = \sqrt{200 - 200\cos \theta}\)

\(AB = \sqrt{200 - 200\cos 2.11} \approx 17.4 \text{ cm}\)