(i) The area of the segment on AP is given by \(\frac{1}{2} r^2 (x - \sin x)\) and the area of the segment on BP is \(\frac{1}{2} r^2 (\pi - x + \sin x)\). Given that the area of the segment on AP is half of the area of the segment on BP, we have:

\(\frac{1}{2} r^2 (x - \sin x) = \frac{1}{2} \times \frac{1}{2} r^2 (\pi - x + \sin x)\)

\(x - \sin x = \frac{1}{2} (\pi - x + \sin x)\)

\(2(x - \sin x) = \pi - x + \sin x\)

\(3x = \pi + \sin x\)

\(x = \frac{1}{3}(\pi + \sin x)\)

(ii) To verify that x lies between 1 and 1.5, calculate:

For \(x = 1\), \(\frac{1}{3}(\pi + \sin 1) \approx 1.380\)

For \(x = 1.5\), \(\frac{1}{3}(\pi + \sin 1.5) \approx 1.497\)

Since \(1.380 < 1.5\) and \(1.497 > 1\), x lies between 1 and 1.5.

(iii) Using the iterative formula \(x_{n+1} = \frac{1}{3}(\pi + \sin x_n)\):

Start with \(x_0 = 1\).

\(x_1 = \frac{1}{3}(\pi + \sin 1) = 1.38052\)

\(x_2 = \frac{1}{3}(\pi + \sin 1.38052) = 1.37492\)

\(x_3 = \frac{1}{3}(\pi + \sin 1.37492) = 1.37417\)

\(x_4 = \frac{1}{3}(\pi + \sin 1.37417) = 1.37405\)

\(x_5 = \frac{1}{3}(\pi + \sin 1.37405) = 1.37403\)

The value of x correct to 3 decimal places is 1.374.