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Math Help - Finding the Centroid of this triangle - stuck!

  1. #1
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    Finding the Centroid of this triangle - stuck!

    I have a triangle CDE

    The coordinates for the vertices are C(-2,4) D(6,2) and E(-4,-2)

    I have found the mid points of CD (2,3) and CE (-3,1)

    I have also found the equation of the median for FE: Y = 5x/6 + 4/3 and GD: Y = x/9 + 4/3

    The problem is that I can't solve the system of equations:

    1. y= 5x/6 + 4/3
    2. y= x/9 + 4/3

    I've tried substituting 1. in to 2. but I end up with:
    x/9 + 4/3 = 5x/6 + 4/3

    Does anyone know where I've gone wrong?
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Finding the Centroid of this triangle - stuck!

    If you want to solve the sytem, substitution is a good solution, you'll get:
    \frac{x}{9}+\frac{4}{3}=\frac{5x}{6}+\frac{4}{3}
    \Leftrightarrow \frac{x}{9}-\frac{5x}{6}=\frac{4}{3}-\frac{4}{3}
    \Leftrightarrow \frac{x}{9}-\frac{5x}{6}=0

    Get a common denominator and a fraction becomes zero if the numerator because zero ...

    Notice if the three vertices of a triangle ABC are A(x_1,y_1),B(x_2,y_2),C(x_3,y_3) then the coordinates of the centroid of the triangle is:
    \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3} \right)
    Last edited by Siron; August 5th 2011 at 02:00 PM.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Finding the Centroid of this triangle - stuck!

    You should know also that:

    In triangle \text{ABC}, with \text{A}=(x_1,y_1), \text{B}=(x_2,y_2), \text{C}=(x_3,y_3).

    The centroid \text{M} is given by:

    \text{M}=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2  +y_3}{3}\right).


    Try to prove the above using the same arguments that you use with your problem(just replace the coordinates)


    Edit:

    Sorry @Siron, didn't see your edited post.
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  4. #4
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    Re: Finding the Centroid of this triangle - stuck!

    Thank you for the replies. I came back here with the problem solved however.

    I realized that I could multiply each number by 18 since 18 is a common denominator of \frac{x}{9}+\frac{4}{3}=\frac{5x}{6}+\frac{4}{3}

    When I did this, I ended up with the correct Centroid. I proved it by making a model in Geometers Sketch Pad and I ended up with the same centroid as the model.

    Thanks!
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Finding the Centroid of this triangle - stuck!

    You're welcome!

    @ Also sprach Zarathrusta:
    No problem
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