Do not use a calculator in answering this question.
\(ABCD\) is a trapezium where \(AB\) is parallel to \(DC\). The area of the trapezium is \(12+11\sqrt3\text{ cm}^2\). \(AB=8-\sqrt{48}\text{ cm}\) and \(DC=12-\sqrt{12}\text{ cm}\).
Find the perpendicular distance between \(AB\) and \(DC\), leaving your answer in the form \(a+b\sqrt3\), where \(a\) and \(b\) are integers.[5]
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