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

### 4 Answers

- PinkgreenLv 73 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.

- oubaasLv 73 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

- oldschoolLv 73 weeks ago
Because it has uniform density. Remember that the mass would balance on the head of a pin at that point.

- JOHNLv 73 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.