Problem Statement
Two ambulance hubs are positioned at coordinates $A(2, 6)$ and $B(8, 2)$ on a regional emergency dispatch grid.
**(a)** [3 marks] Find the equation of the perpendicular bisector of the line segment $[AB]$ in the form $y = mx + c$.
**(b)** [2 marks] An emergency occurs at point $P(5, 5)$. By computing distances or using your diagram, state which hub will respond to the incident.
Verified Solution & Marking Scheme
Part (a) [M1]
Midpoint of $AB = \left(\frac{2+8}{2}, \frac{6+2}{2}\right) = (5, 4)$
Part (a) [M1]
Gradient of $AB = \frac{2 - 6}{8 - 2} = \frac{-4}{6} = -\frac{2}{3} \implies m_{\perp} = \frac{3}{2}$
Part (a) [A1]
$y - 4 = \frac{3}{2}(x - 5) \implies y = \frac{3}{2}x - \frac{7}{2}$
Part (b) [M1]
$d(P, A) = \sqrt{(5-2)^2 + (5-6)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16$
Part (b) [A1]
$d(P, B) = \sqrt{(5-8)^2 + (5-2)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24$. Ambulance Hub A will respond.