Problem E

Diagonal Distance

Square \(ABCD\) has \(K\) on \(BC\), \(L\) on \(DC\), \(M\) on \(AD\), and \(N\) on \(AB\) such that \(KLMN\) forms a rectangle, \(\triangle AMN\) and \(\triangle LKC\) are congruent isosceles triangles, and also \(\triangle MDL\) and \(\triangle BNK\) are congruent isosceles triangles. If the total area of the four triangles is \(50\) cm\(^2\), what is the length of \(MK\)?