Problem Statement
Find the equation of the perpendicular bisector of the line segment joining the points $A(4, 2)$ and $B(-2, 6)$.
Verified Solution & Marking Scheme
Find Midpoint M of AB
$$M = \left(\frac{4 + (-2)}{2}, \frac{2 + 6}{2}\right) = \left(\frac{2}{2}, \frac{8}{2}\right) = (1, 4)$$
Find Slope of AB and Perpendicular Slope
$$m_{AB} = \frac{6 - 2}{-2 - 4} = \frac{4}{-6} = -\frac{2}{3}$$
Since the perpendicular bisector is perpendicular to $AB$:
$$m_\perp = -\frac{1}{m_{AB}} = -\frac{1}{-2/3} = \frac{3}{2}$$
Write Equation of Line Through M(1, 4)
$$y - 4 = \frac{3}{2}(x - 1) \implies 2(y - 4) = 3(x - 1)$$
$$2y - 8 = 3x - 3 \implies 3x - 2y + 5 = 0$$