Why would the centroid of a triangle also be its center of mass?

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  • 3 weeks ago

    ..............A

    ............/.*.\

    ........../...*...\

    ......../.....*.....\

    ....B/......M......\C

    ..../........ *........\.

    ../...........*..........\

    D...........F...........E

    Consider the triangle ADE, F is the mid-point

    of the side DE=>AF is the median. M is a point

    at the intersection of line-segment BC & AF.

    Tri. AMB~ Tri. AFD=>

    AM/AF=MB/FD=>FD=AF*MB/AM

    Similarly, FE=AF*MC/AM.

    FD=FE=>AF*MB/AM=AF*MC/AM=>MB=MC

    => M is also a mid-point of BC. If Tri.ADE is

    made of some material, then the weight of

    Tri. ADF=that of Tri. AEF.

    Similarly, M is a point on the median from AD

    to E. Also M is a point on the median from AE

    to D=> M is the intersection point of the 3 medians; i.e. the centroid =the center of mass.

  • oubaas
    Lv 7
    3 weeks ago

    Yesssssssssir :

    Centroid coordinates = i(x1+x2+x3)/3 ; j(y1+y2+y3)/3

    C. o. M. coordinates = i(x1*m1+x2*m2+x3*m3)/(3*(m1+m2+m3)) ; j(y1*m1+y2*m2+y3*m3)/(3*(m1+m2+m3))

    since a geometrical figure is homogeneous allover its surface , then m1 = m2 = m3 = m , then :

    C. o. M. = m*i*(x1+x2+x3)/(3*m) ; m*j(y1+y2+y3)/(3m)

    mass m cross

    C. o. M. = i*(x1*m1+x2*m2+x3*m3)/3 ; j*(y1*m1+y2*m2+y3*m3)/3...which is exactly the formula of the centroid

  • 3 weeks ago

    Because it has uniform density. Remember that the mass would balance on the head of a pin at that point.

  • JOHN
    Lv 7
    3 weeks ago

    The reason why the place where the 3 medians meet is called the centroid is that this point also happens to be the centre of mass. And the reason why the centre of mass is at the centroid is that that is the way the maths works out. I am not going to give a proof of this latter fact because this doesn’t seem to be what is puzzling you.

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