(i) Find the equation of the perpendicular bisector of AB.
1. Calculate the midpoint M of AB:
\(M = \left( \frac{4 + 10}{2}, \frac{6 + 2}{2} \right) = (7, 4)\)
2. Find the slope of AB:
\(m_{AB} = \frac{2 - 6}{10 - 4} = -\frac{2}{3}\)
3. The slope of the perpendicular bisector is the negative reciprocal:
\(m_{\text{perpendicular}} = \frac{3}{2}\)
4. Use point-slope form to find the equation:
\(y - 4 = \frac{3}{2}(x - 7)\)
(ii) Calculate the coordinates of C.
1. Equation of line parallel to AB through (3, 11):
\(y - 11 = -\frac{2}{3}(x - 3)\)
2. Solve the system of equations:
Equation 1: \(y - 4 = \frac{3}{2}(x - 7)\)
Equation 2: \(y - 11 = -\frac{2}{3}(x - 3)\)
3. Substitute and solve:
\(\frac{3}{2}(x - 7) + 4 = -\frac{2}{3}(x - 3) + 11\)
4. Solve for x:
\(\frac{3}{2}x - \frac{21}{2} + 4 = -\frac{2}{3}x + 2 + 11\)
5. Simplify and solve:
\(\frac{3}{2}x + \frac{2}{3}x = \frac{21}{2} - 4 + 13\)
6. Find x:
\(x = 9\)
7. Substitute x back to find y:
\(y - 4 = \frac{3}{2}(9 - 7)\)\(y = 7\)
Coordinates of C: (9, 7)