1.19: Past-paper question 2

The point $\displaystyle A$ has coordinates $\displaystyle (3, 2)$, the point $\displaystyle B$ has coordinates $\displaystyle (8, 3)$ and the point $\displaystyle C$ has coordinates $\displaystyle (4, 7)$

(a) Show that $\displaystyle ABC$ is an isosceles triangle.

The midpoint of $\displaystyle BC$ is $\displaystyle M$

(b) Find an equation of the line that passes through $\displaystyle A$ and $\displaystyle M$

Give your answer in the form $\displaystyle y = mx + c$

The points $\displaystyle A$, $\displaystyle C$ and $\displaystyle D$ are collinear such that $\displaystyle AD = kAC$ ($\displaystyle k > 1$)

Given that $\displaystyle \angle ABD = 90^\circ$

(c) find the coordinates of $\displaystyle D$

Show your working clearly.