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Math Help - Perimeter of Triangle

  1. #1
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    Perimeter of Triangle

    Given a point (a,b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a,b), one on the x-axis, and one on the line y=x.

    Thanks!
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  2. #2
    Super Member

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    Hello, RubyRed!

    Who assigned this? . . . It is an ugly problem!


    Given a point P(a,b) with 0 < b < a,
    determine the minimum perimeter of a triangle with one vertex at P,
    one on the x-axis, and one on the line y=x
    Code:
            |                 *
            |               *
            |           R *
            |      (y,y)o
            |         *  *  *
            |       *     *     *    P
            |     *        *        o(a,b)
            |   *           *     *
            | *              *  *
          - + - - - - - - - - o - - - - -
            |               (x,0)
            |                 Q

    Let Q(x,0) be on the x-axis.

    Let R(y,y) be on the line y = x


    Then: . \text{Perimeter} \;=\;\overline{PQ} + \overline{QR} + \overline{RP}

    . . P \;=\;\bigg[(x-a)^2 + b^2\bigg]^{\frac{1}{2}} + \bigg[(x-y)^2 + y^2\bigg]^{\frac{1}{2}} + \bigg[(y-a)^2 + (y-b)^2\bigg]^{\frac{1}{2}}

    . . P \;=\;\bigg[x^2-2ax + a^2+b^2\bigg]^{\frac{1}{2}} + \bigg[x^2-2xy + 2y^2\bigg]^{\frac{1}{2}} + \bigg[2y^2 - 2(a+b)y + (a^2+b^2)\bigg]^{\frac{1}{2}}


    And now solve the system: . \begin{Bmatrix}\dfrac{\partial P}{\partial x} &=& 0 \\ \\[-3mm] \dfrac{\partial P}{\partial y} &=& 0 \end{Bmatrix}
    Good luck!

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