Problem Statement
Three mobile network cell towers are located at $A(1, 2)$, $B(5, 2)$, and $C(3, 6)$.
**(a)** [3 marks] Find the equation of the perpendicular bisector of the line segment $[AB]$.
**(b)** [4 marks] Find the coordinates of the Voronoi vertex equidistant from towers $A$, $B$, and $C$.
Verified Solution & Marking Scheme
Part (a) [M1]
Midpoint of $[AB]$: $M_{AB} = \left( \frac{1+5}{2}, \frac{2+2}{2} \right) = (3, 2)$
Part (a) [M1]
Line segment $[AB]$ is horizontal with gradient $0$.
Part (a) [A1]
Perpendicular bisector is the vertical line $x = 3$.
Part (b) [M1]
Find perpendicular bisector of $[AC]$: midpoint $M_{AC} = (2, 4)$, gradient $m_{AC} = \frac{6-2}{3-1} = 2$
Part (b) [M1]
Perpendicular gradient $m_\perp = -\frac{1}{2}$. Line: $y - 4 = -\frac{1}{2}(x - 2) \implies y = -\frac{1}{2}x + 5$
Part (b) [M1]
Substitute $x = 3$ into $y = -\frac{1}{2}(3) + 5 = 3.5$
Part (b) [A1]
The Voronoi vertex is at $(3, 3.5)$.