prove that the medians of a triangle are concurrent and find the point of concurrency . also show that this point divides the medians in the ratio 2:1 . what is he point known as?

Dear Student,
Please find below the solution to the asked query:



Reflect the triangle along AC, you can get a diagram above.Here G is the centroid.ABCB1  is a parallelogram. BEB1  is a straight line.Since  CD = AD1  and  CDAD1, DCD1A  is a parallelogram.  Opposite sides equal and parallel.Hence DGCG1  Since  BD = DC and DGCGHence in BCG1, by Basic Proportonality theorem we get:BG=GG1BG:GG1=1:1As GE=EG1BG:GE=2:1Hence centroid divides median in 2:1.This point G is called centroid of the triangle.

As P is the mid point of NO, hence co-ordinates of P will be Px2+x32,y2+y32.As G1 divides MP in the ratio 2:1, hence by section formula co-ordinates of G1 will be:G11.x1+2x2+x321+2,1.y1+2y2+y321+2=G1x1+x2+x33,y1+y2+y33As Q is the mid point of MO, hence co-ordinates of Q will be Qx1+x32,y1+y32Let G2 be a point on median NQ which divides it in the ratio 2:1., henceby section formula co-ordinates of G2 will be:G21.x2+2x1+x321+2,1.y2+2y1+y321+2=G2x1+x2+x33,y1+y2+y33As R is the mid point of MN, hence co-ordinates of R will be Rx1+x22,y1+y22Let G3 be a point on median OR which divides it in the ratio 2:1., henceby section formula co-ordinates of G3 will be:G31.x3+2x1+x221+2,1.y3+2y1+y221+2=G3x1+x2+x33,y1+y2+y33As co-ordinates of G1,G2,G3 are same, hence medians of triangle are concurrent.

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards

  • 1
What are you looking for?