Problem Statement
Find the equation of the perpendicular bisector of the line segment joining the points $A(3, -2)$ and $B(-1, 4)$.
Verified Solution & Marking Scheme
Find midpoint M of AB
$$M(x, y) = \left( \frac{3 + (-1)}{2}, \frac{-2 + 4}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)$$
Calculate slope of AB and perpendicular slope
$$\text{Slope of } AB (m_1) = \frac{4 - (-2)}{-1 - 3} = \frac{6}{-4} = -\frac{3}{2}$$
Since the bisector is perpendicular to $AB$, its slope $m_2 = -\frac{1}{m_1} = \frac{2}{3}$.
Find equation using point-slope form through M(1, 1)
$$y - y_1 = m_2(x - x_1) \implies y - 1 = \frac{2}{3}(x - 1)$$
$$3(y - 1) = 2(x - 1) \implies 3y - 3 = 2x - 2 \implies 2x - 3y + 1 = 0$$