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Math Help - Minimize perimeter

  1. #1
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    Minimize perimeter

    An isosceles triangle is circumscribed about a circle of radius R. Express the perimeter of the triangle as a function of its altitude, and show that the perimeter is a minimum when this altitude is 3R.

    This should not be that hard, but I cannot for the life of me figure out the equation in terms of the altitude and radius ONLY. I have too many variables. Thanks for you help.
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  2. #2
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    This was kind of tricky. I made a diagram.

    By the included diagram: R=OB=OC; y=AD, h=AB

    AB is the height of the triangle, y is the length of the side.

    Look at triangle ABC. h^{2}+x^{2}=y^{2}

    Look at triangle AOC. (h-R)^{2}=R^{2}+(y-x)^{2}

    Using these, we can solve for x. x=\frac{Rh}{\sqrt{h^{2}-2hR}}

    The area of a triangle is 1/2xh.

    \frac{1}{2}\frac{Rh^{2}}{\sqrt{h^{2}-2hR}}

    There, R is a constant and it is in terms of h. Now, differentiate:

    \frac{dA}{dh}=\frac{Rh(h-3R)}{(h-2R)\sqrt{h(h-2R)}}

    Set to 0 and solve for h.

    Rh(h-3R)=0

    h=3R
    Last edited by galactus; November 24th 2008 at 05:38 AM.
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  3. #3
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    3 things:

    you assumed cd=db.

    The problem wants perimeter minimized.

    I don't think that solution is correct
    Last edited by jabroni1212; November 29th 2007 at 04:34 PM.
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