(i) To find the equation of the perpendicular bisector of \(AB\):

1. Calculate the midpoint of \(AB\):

\(\left( \frac{5+9}{2}, \frac{7+(-1)}{2} \right) = (7, 3)\)

2. Find the slope of \(AB\):

\(m = \frac{-1 - 7}{9 - 5} = -2\)

3. The slope of the perpendicular bisector is the negative reciprocal:

\(m_{\text{perp}} = \frac{1}{2}\)

4. Use the point-slope form to find the equation:

\(y - 3 = \frac{1}{2}(x - 7)\)

(ii) To find the distance \(BX\):

1. Find the equation of the line through \(C(1, 2)\) parallel to \(AB\):

\(y - 2 = -2(x - 1)\)

2. Solve the system of equations:

\(\begin{align*} y - 3 &= \frac{1}{2}(x - 7) \\ y - 2 &= -2(x - 1) \end{align*}\)

3. Substitute and solve:

\(\frac{1}{2}x - \frac{1}{2} = -2x + 4\)

\(x = \frac{9}{5}, \quad y = \frac{2}{5}\)

4. Calculate the distance \(BX\):

\(BX = \sqrt{(9 - \frac{9}{5})^2 + (-1 - \frac{2}{5})^2}\)

\(BX = \sqrt{7.2^2 + 1.4^2}\)

\(BX \approx 7.33\)