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Homework Help: Center mass of triangle in 3-D space

  1. Sep 6, 2011 #1
    1. The problem statement, all variables and given/known data
    So I have a triangle with points: A (4,2,0), B (3,3,0), and C (1,1,3). We are to find the point at which the three medians intersect in i,j,k format. I've found the midpoints of each side but I don't know where to go from there.
     
  2. jcsd
  3. Sep 6, 2011 #2
    Construct the equation of the line joining the midpoint and the opposite vertex. Do this for all vertices.
     
  4. Sep 6, 2011 #3

    dynamicsolo

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    You can save a good bit of this work if you are allowed to use the theorem which states that the medians of a triangle intersect at a point with divides each median in the ratio of 2:1 . Then you can choose any one median and find the point (by proportions) which is one-third of the way from the base to the opposite vertex. (Any other choice of a median to work with should give exactly the same result.)

    I am presuming in this that the problem is not asking you to show that all three medians meet at the point with this property.
     
  5. Sep 6, 2011 #4

    HallsofIvy

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    The simplest way to do this: the coordinates of the centroid of a triangle (not, strictly speaking, the "center of mass" because a geometric figure does not have 'mass') is the mean of the coordinates of the three vertices. That is, if the vertices of the triangle are at [itex](x_1, y_1, z_1)[/itex], [itex](x_2, y_2, z_2)[itex], [itex](x_3, y_3, z_3)[/itex], then the centroid is at
    [tex]\left(\frac{x_1+ x_2+ x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)[/tex]

    If you don't have that theorem, use that as a check.
    (The average of the coordinates works for the two dimensional triangle or three dimensional tetrahedron but not for other figures.)
     
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