MathematicsThe maximum area of the rectangle whose sides pass through the vertices of a given rectangle of sides a and b is

The maximum area of the rectangle whose sides pass through the vertices of a given rectangle of sides a and b is

Area, $A = (a \sin θ + b \cos θ) (a \cos θ + b \sin θ)$

$= a b + \frac{(a^{2} + b^{2})}{2} \sin 2 θ$

A is maximum when sin $2 θ$ is maximum.

Therefore, $A_{max} = a b + \frac{(a^{2} + b^{2})}{2} = \frac{1}{2} (a + b)^{2}$

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