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    Prove that the rectangle with greatest area that can be inscribed in a circle is a square and find an expression for the length of the sides of this square.
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    (Original post by Ano123)
    Prove that the rectangle that can be inscribed in a circle with greatest area is a square and find an expression for the length of the sides of this square.
    What have you tried?

    Start with a circle of radius r and centre O. Inscribe a rectangle with lengths a and b.

    The area of the rectangle is A=ab. Express a and b in terms of r and \theta from cosine rule if you let theta equal the angle between ON and OM where NM=a and ON=OM=r. Same idea for the b side.

    Then once you have A in terms of r and theta, differentiate A with respect to theta and maximise it by making it equal to 0. You should find the angle which makes the area max, and apply it to a and b. You should find that a=b in terms of r therefore it's a square.
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    (Original post by Ano123)
    Prove that the rectangle with greatest area that can be inscribed in a circle is a square and find an expression for the length of the sides of this square.
    By drawing a good diagram, you should be able to see that the radius of the circle equals half the diagonal of the rectangle. Now you can complete the problem.
 
 
 
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Updated: September 10, 2016
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