Problem D

Find the Largest Area

Rectangle \(ACEG\) has \(B\) on \(AC\) and \(F\) on \(EG\) such that \(BF\) is parallel to \(CE\). Also, \(D\) is on \(CE\) and \(H\) is on \(AG\) such that \(HD\) is parallel to \(AC\), and \(BF\) intersects \(HD\) at \(J\). The area of rectangle \(ABJH\) is \(6\text{ cm}^2\) and the area of rectangle \(JDEF\) is \(15\text{ cm}^2\).

If the dimensions of rectangles \(ABJH\) and \(JDEF\), in centimetres, are integers, then determine the largest possible area of rectangle \(ACEG\). Note that the diagram is just an illustration and is not intended to be to scale.