A square and a rectangle have the same perimeter p and their areas differ by q units. What is the square of the difference between the length and breadth of the rectangle ?

Explanation

Let the square have side a, so perimeter p = 4a. For the rectangle, 2(l+b) = 4a \implies a = \frac{l+b}{2}. Area difference q = a^2 - lb = (\frac{l+b}{2})^2 - lb = \frac{l^2 + b^2 + 2lb - 4lb}{4} = \frac{(l-b)^2}{4}. Thus, (l-b)^2 = 4q.

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