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# Circles

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1. 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.
2. (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?

The area of the rectangle is . Express a and b in terms of and 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.
3. (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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