show that all the rectangles of maximum perimeter which can be inscribed in a circle of radius 'a' is square of side root2a

Dear girvinder,

The largest rectangle would be a square. If the circle has radius a, the diameter is 2a. This diameter would also be the diameter of a square of side length b. Using the Pythagorean theorem, b² + b² = (2a)².
2b² = 4a²
b² = 2a²
b = √(2a²) or a√2 = the length of the sides of the square
The area of a square of side length b is therefore (√(2a²))² = 2a² which is the largest area for a rectangle inscribed in a circle of radius a.

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