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Math Help - Medians of a triangle

  1. #1
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    Medians of a triangle

    Can someone help me please? This is very confusing. =S

    Prove that the three medians of a triangle are concurrent by choosing as vertices A(6a,6b) B(-6a-6b) and C(0,6c) and following these steps:

    a) Find the Midpoints P,Q and R of BC, CA and AB respectively. Show that the median through C is x=0 and find the equation of the other two medians.

    b) Find where the median through C meets another median, and show that the point lies on the third median.

    Thanks
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  2. #2
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    Hello deltaxray
    Quote Originally Posted by deltaxray View Post
    Can someone help me please? This is very confusing. =S

    Prove that the three medians of a triangle are concurrent by choosing as vertices A(6a,6b) B(-6a-6b) and C(0,6c) and following these steps:

    a) Find the Midpoints P,Q and R of BC, CA and AB respectively. Show that the median through C is x=0 and find the equation of the other two medians.

    b) Find where the median through C meets another median, and show that the point lies on the third median.

    Thanks
    You need to know:

    • the mid-point of the line joining (x_1,y_1) to (x_2,y_2) is \Big(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\Big)


    • the equation of the line joining (x_1,y_1) to (x_2,y_2) is y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)

    Use these results to find the mid-points and the equations of the three medians. Then solve these equations simultaneously to find their point(s) of intersection.

    Let us know if you can't complete it now.

    Grandad
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  3. #3
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    If my calculations are correct....

    R = (0, 0)
    P = (-3a, 3c - 3b)
    Q = (3a, 3c + 3b)

    I did not try to work out the equations for AB, BC, AC in terms of a, b, c, x, y.......they will look too messy.

    median RC is x = 0 so lies on y-axis. if any other medians, PA or BQ, is to intersect with RC, the x coordinate of the intersection point must be 0.

    let AP intersects with RC at (0, m1).
    (6b - m1)/(6a - 0) = (3c - 3b - m1)/(-3a - 0)
    in the end i get m1 = 2c

    by the same method for the interection between BQ and RC....say they intersect at (0, m2).....i also get m2 = 2c in the end.

    Hence the three medians PA, BQ, RC all intersect at (0, 2c)
    Last edited by ukorov; November 1st 2009 at 04:59 AM.
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