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Math Help - Find the area of the largest isosceles triangle that can be inscribed in a circulus.

  1. #1
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    Find the area of the largest isosceles triangle that can be inscribed in a circulus.

    I have this problem from Calculus 1 that I need help with. I have no idea where to start.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Find the area of the largest isosceles triangle that can be inscribed in a circul

    I would use a bit of symmetry here. Bisect the circle and the isosceles triangle so that you have a right triangle within a semi-circle. Let the semi-circle be described by:

    y=\sqrt{4^2-x^2}

    Let the hypotenuse of the right triangle extend from (-4,0) to (x,y) and the horizontal leg from (-4,0) to (x,0) and the vertical leg from (x,y) to (x,0).

    The area A of the triangle is then:

    A=\frac{1}{2}bh=\frac{1}{2}(x+4)(\sqrt{4^2-x^2})

    Now, maximize this function, then double the maximum area to find the maximum area for the original problem.
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