If \(AB=BC\), \(AB=a\), \(DE=b\), and \(\angle ABC =\angle EDC = 90^{\circ}\) then which of the following is an expression for the area of quadrilateral \(ABDE?\)

A. \(\large\frac{a^2}{2}-\frac{b^2}{2}\)

B. \(\large\frac{a^2}{2}+\frac{b^2}{2}\)

C. \(\large\frac{a^2}{4}-\frac{ab}{2}\)

D. \(\large a^2-b^2\)

E. \(\large\frac{a^2}{4}+\frac{ab}{2}\)

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