Part (i):

The slope of line \(AB\) is \(m = \frac{3 - 7}{8 - 0} = -\frac{1}{2}\).

The equation of line \(AB\) is \(y = -\frac{1}{2}x + 7\).

Substitute \(C(3k, k)\) into the line equation:

\(k = -\frac{1}{2}(3k) + 7\)

\(k = -\frac{3}{2}k + 7\)

\(2k = -3k + 14\)

\(5k = 14\)

\(k = 2.8\)

Part (ii):

The midpoint of \(AB\) is \(M(4, 5)\).

The perpendicular slope is \(2\), so the equation of the perpendicular bisector is:

\(y - 5 = 2(x - 4)\)

\(y = 2x - 3\)

Substitute \(C(3k, k)\) into the bisector equation:

\(k = 2(3k) - 3\)

\(k = 6k - 3\)

\(5k = 3\)

\(k = 0.6\)