(i) To find the perimeter of the shaded region, we first use the Pythagorean theorem in triangle OPT to find OT:

\(OT = \sqrt{OP^2 + PT^2} = \sqrt{5^2 + 12^2} = 13 \text{ cm}\)

Next, find QT:

\(QT = OT - OQ = 13 - 5 = 8 \text{ cm}\)

Calculate the angle POQ using the tangent function:

\(\angle POQ = \tan^{-1}\left(\frac{12}{5}\right) = 1.176 \text{ radians}\)

Find the arc length S:

\(S = r\theta = 5 \times 1.176 = 5.88 \text{ cm}\)

Thus, the perimeter of the shaded region is:

\(5.88 + 12 + 8 = 25.9 \text{ cm}\)

(ii) To find the area of the shaded region, calculate the area of sector POQ:

\(\text{Area of sector} = \frac{1}{2}r^2\theta = \frac{1}{2} \times 5^2 \times 1.176 = 14.7 \text{ cm}^2\)

Calculate the area of triangle OPT:

\(\text{Area of triangle} = \frac{1}{2} \times 12 \times 5 = 30 \text{ cm}^2\)

The area of the shaded region is:

\(30 - 14.7 = 15.3 \text{ cm}^2\)