Areas Inside a Rectangle

Geometry Level 4

\(ABCD\) is a rectangle with point \(E\) on line segment \(AB\) and point \(F\) on line segment \(BC\) such that \([ADE] = 7\), \([BEF] = 11\) and \([DCF] = 9\). Then \([DEF] = a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is not a multiple of the square of any prime. What is the value of \(a + b\)?

Details and assumptions

\([PQRS]\) denotes the area of figure \(PQRS\).

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