2026 NORTHBROOKS P1 Q8

The diagram shows a quadrilateral \(ABCD\). Point \(B\) is \((8,3)\), \(C\) is \((3k,5k+3)\), \(D\) is \((-2,5)\) and \(A\) lies on the \(x\)-axis. The equation of \(AB\) is \(5y=3x-9\) and angle \(ADC=90^\circ\).

1. Find the coordinates of \(A\).[1]
2. Show that \(k=2\).[4]
3. Find the equation of the perpendicular bisector of \(DC\).[3]
4. Find the area of the quadrilateral \(ABCD\).[2]