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Math Help - Calculating the base of a triangle, with median and legs

  1. #1
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    Thumbs up Calculating the base of a triangle, with median and legs

    Hi Forum,

    I searched in some places and I couldn't find a good way to do this. The way the solution is presented is not very clear.

    In the triangle ABC, AB=4 and AC=8. If M is the mid point of BC and AM=3 what is the length of BC?

    Code:
         
            A
    4                    8
               3        
          x           y
    B          M           C
    Now, if M is the midpoint between B and C then x and y are equal.
    The solution considers them as two different numbers.
    AM is the median, right?
    How should I proceed now?

    Thanks!
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by Zellator View Post
    Hi Forum,

    I searched in some places and I couldn't find a good way to do this. The way the solution is presented is not very clear.

    In the triangle ABC, AB=4 and AC=8. If M is the mid point of BC and AM=3 what is the length of BC?

    Code:
         
            A
    4                    8
               3        
          x           y
    B          M           C
    Now, if M is the midpoint between B and C then x and y are equal.
    The solution considers them as two different numbers.
    AM is the median, right?
    How should I proceed now?

    Thanks!
    1. Draw parallels to AB and AC respectively through M. The parallels divide the sides of the triangle in two equal parts. Together with the known sides of the triangle you'll get a parallelogram with one known diagonal.

    2. Use Cosine rule to determine the value of the angle \alpha (see attachment)

    3. Determine the value of the angle \gamma:

    |\gamma| = 180^\circ - |\alpha|

    4. Use the Cosine rule and the angle \gamma to determine the length of BC. You should come out with |\overline{BC}|\approx 11.4898
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  3. #3
    MHF Contributor
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    It's also possible to use this formula.
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  4. #4
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    Thanks earboth and emakarov!
    I searched before posting this question, even in wikipedia.
    I guess I overlooked it. It's not so usual to calculate something like this, is it?

    Thanks for your help.
    I like to see different points of view, earboth explanation is really clever!
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  5. #5
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    if the fourth vertice D is belowBC only the cosine of angle DAC is required to solve for x= to 1/2 BC. The angle does not have to be found.



    bjh
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