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Math Help - "Find point C on the X-axis so that AC+BC is a minimum"

  1. #1
    Junior Member
    Joined
    Aug 2007
    Posts
    54

    "Find point C on the X-axis so that AC+BC is a minimum"

    These are just thrown into an assignment on Reflections, and I for one don't get it.

    We've got about 20 problems of them, so I can't really just try and hope it's right, either.

    Examples:

    A(1,4) B(4,2)

    A(-2,3) B(6,3)

    A(-3,2) B(-6,4)

    A(1,-3) B(-3,-3)

    I'm not even sure what it's asking for... Am I supposed to find a new point that would create the shortest possible line segments if graphed, or what?

    If it's something else, what formulae would I use?
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Hi

    Yes you are supposed to find a new point that would create the shortest sum of the distances AC+BC, C being on X-axis

    The solution is given in the sketch below
    A' being the symmetric point of A with respect to X-axis
    A'C=AC
    AC+BC=A'C+BC
    This latter distance is shortest when A',B and C are aligned

    This problem is also known as "the problem of the river" : suppose that you are in A and you want to go to B after getting some water at the river (X-axis), where do you have to get the water in order to minimize the global distance ?

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