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

### 3 Answers

- PinkgreenLv 710 months 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 710 months 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 710 months ago
Because it has uniform density. Remember that the mass would balance on the head of a pin at that point.