The coordinates of a triangle `ABC` are `A(1,2)`, `B(4,6)` and `C(-3,-2)`. Three particles of masses `1kg`, `2kg` and `m kg` are placed the vertices of the triangle. If the coordinate of the centre of mass are `((3)/(5),2)`, calculate the mass `m`.

Explanation

`x_(cm)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(m_(1)+m_(2)+m_(3))=(3)/(5)`
`rArr (1+8-3m)/(3+m)=(3)/(5)`
`rArr (9-3m)/(3+m)=(3)/(5)`
`rArr 45 - 15m = 9+3m`
`rArr 36=18 m`
`rArr m = 2kg`
Or
Alternate method
`y_(cm)=(m_(1)y_(1)+m_(2)y_(2)+m_(3)y_(3))/(m_(1)+m_(2)+m_(3))=2`
`rArr (2+12-2m)/(3+m)=2`
`rArr 14-2m=6+2m`
`rArr 8=4m`
m = 2 kg.