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Math Help - Simple problem, We know this but how to prove this?

  1. #1
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    Simple problem, We know this but how to prove this?

    I had a question in my exam.
    Code:
    Prove that a line segment has only one mid point.
    I wrote that its not possible to have more than one mid point as it has to be equal on both sides. i got 1 mark outta 3 marks. Can anyone gimme proper answer with proof and all?
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  2. #2
    MHF Contributor
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    You have only suggested that there must be only one. You have not proven it.

    1) Draw a line segment with a midpoint.
    2) Note that the two sides have identical length
    3) Put another midpoint on the line segment. I doesn't matter where.
    4) Note that the distance between the two midpoints measures other than zero.
    5) Call the three segments so defined A, B, and C. B is between the two midpoints.
    6) If both are midpoints, then mA = mB + mC AND mA + mB = mC
    7) What does this say about the length of B?
    8) What does that say about where the second "midpoint" really is?
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  3. #3
    Junior Member
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    Hmm...would you be allowed to use perpendicular bisectors?

    This is how I would approach it

    All points equidistant from endpoints A and B would be on the perpendicular bisector, however, only one point, the midpoint, is on the perpendicular bisector and \overline{AB}. Therefore exactly one midpoint exists on a line segment.
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