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Finding the centroid of the triangular region

  1. Mar 26, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the coordinates of the centroid G of the triangular region with vertices (0,0),(a,0),(b,c).

    2. Relevant equations

    for the centroid x = (1 / area) * double integral ( x dA)
    y = (1 / area) * double integral ( y dA)

    3. The attempt at a solution

    Ok, what i did so far for this was try to get an equation for the lines on the left and right of the triangle. i got x = by/c and x = -y(a-b)/c + a (both were found using point slope)

    Then I integrated with respects to x first and used the above equations as my limits of integration, and then integrated with respects to y and used 0 and c as limits of integration.

    I want to know if that sounds like the right method of going into this problem. I get a really long mess of a's and b's for x and c canceled out. It feels like I'm missing something.

    The whole point of the assignment was to try to prove that the three medians of a triangle intersect the centroid, but if I'm already going in the right direction, I'm sure I can figure the rest out.
     
  2. jcsd
  3. Mar 27, 2007 #2
    Without actually seeing the math you did, it looks like your idea of what to do is right.
     
  4. Mar 27, 2007 #3
    centroid is given by coordinates [tex] x= \frac{x_1+x_2+x_3}{3} , \ y= \frac{y_1+y_2+y_3}{3} [/tex]
     
    Last edited: Mar 27, 2007
  5. Mar 27, 2007 #4
    [tex]G ( {x=x_1+x_2+x_3/3} , {y=y_1+y_2+y_3}/3 )[/tex]

    Can be proved by using midpoint theorem and the fact that medians bisect each other in ratio 2:1
     
  6. Mar 27, 2007 #5

    HallsofIvy

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    Of course, that's assuming what the OP was asked to show: that the medians intersect at the centroid.

    Chumatha87, your basic idea is correct. Integrating with respect to x, you will want to divide the integral in two parts: 0 to b and b to a.

    Don't forget to divide by the area which is (1/2)ac.
     
  7. Mar 27, 2007 #6
    coordinates are G(a+b/3,c/3)

    for points (0,0),(a,0),(b,c).
     
  8. Mar 27, 2007 #7

    HallsofIvy

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    All you are telling us is that you do not understand what the original question was.
     
  9. Mar 28, 2007 #8
    alright i got the answer as ( (a+b)/3, c/3 ) for the centroid using double integrals, it seems that I divided by the area in one part of the equation, but neglected to do it in another part, so things didn't cancel out the first time. Thanks for the help guys.
     
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