2024 ACS(BR) P1 Q5

The diagram shows points \(A(5,0)\), \(B(a,3a)\) and \(C(-2,6)\) such that the line \(AB\) is perpendicular to the line \(BC\).

1. Show that \(a=2.5\).[3]
2. Find the coordinates of the midpoint of \(AC\). Hence find the coordinates of \(D\) such that \(ABCD\) is a parallelogram.[3]
3. The area of triangle \(AEC\) is \(5\text{ units}^2\). Find the value of \(x\) given that the point \(E\) is \((x,2x)\), where \(x>1\).[2]