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Math Help - Calculating the Minimum Perimeter of a Triangle

  1. #1
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    Calculating the Minimum Perimeter of a Triangle

    Hey there sorry if this is in the wrong thread, but I need some help with this problem I'm stuck on.

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

    I hope someone can help/guide me through this problem, so I can understand how to do this problem thanks.
    Last edited by gfbrd; October 12th 2012 at 06:14 PM.
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  2. #2
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    Re: Calculating the Minimum Perimeter of a Triangle

    No triangle exists that minimizes that condition.
    However, the infinum of the perimeters of all such triangles = twice the distance from (a,b) to the line y = x.
    That's just an application of the triangle inequality after picking the first two points (one is on the line, the other is (a,b)) of any triangle.
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  3. #3
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    Re: Calculating the Minimum Perimeter of a Triangle

    Hello, gfbrd!

    Did you make a sketch?


    Give 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.

    Code:
          |
          |                     *
          |                   *
          |               R *
          |               o(y,y)
          |             **  *
          |           * *     *
          |         *  *        * P
          |       *   *           o(a,b)
          |     *    *        *
          |   *     *     *
          | *    Q *  *
      - - * - - - o - - - - - - - - - - - -
          |     (x,0)
          |
    Point P has given coordinates P(a,b).
    It lies below the 45o-line.

    Point Q lies on the x-axis with coordinates Q(x,0).

    Point R lies on the line y = x with coordinates R(y,y).


    The lengths of the sides of the triangle are:

    . . PQ \;=\;\sqrt{(x-a)^2 + b^2}

    . . QR \;=\;\sqrt{(x-y)^2 + y^2}

    . . PR \;=\;\sqrt{(y-a)^2 + (y-b)^2}


    The perimeter of the triangle is:

    . . Z \;=\;\sqrt{(x-a)^2 + b^2} + \sqrt{(x-y)^2 + y^2} + \sqrt{(y-a)^2 + (y-b)^2}

    And that is the function we must minimize.
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  4. #4
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    Re: Calculating the Minimum Perimeter of a Triangle

    Apologies - ignore my previous post - it's wrong. I simply read right over the phrase "on the x-axis", and so didn't include that condition.
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