Three particles of masses 1 kg, 2 kg, and 3 kg are located the vertices of an equilateral triangle of side a. The x, y coordinates of the centre of mass are

Let the masses 1 kg,  $2$ kg and 3 kg are located at the vertices A, B and C as shown in figure above. The coordinates of points A, B and C are (0, 0), (a, 0), ($\frac{\mathrm{a}}{2}, \frac{\sqrt{3}\mathrm{a}}{2}$) respectively.

The coordinates of centre of mass are

${\mathrm{X}}_{\mathrm{CM}} = \frac{{\mathrm{m}}_1{\mathrm{x}}_1+{\mathrm{m}}_2{\mathrm{x}}_2+{\mathrm{m}}_3{\mathrm{x}}_3}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3}$

         = $\frac{[1\times 0+{\displaystyle 2}\times \mathrm{a}+3\times {\displaystyle \frac{\mathrm{a}}{2}}]}{[1+{\displaystyle 3}+2]} = \frac{7\mathrm{a}}{12}$

${\mathrm{Y}}_{\mathrm{CM}} = \frac{{\mathrm{m}}_1+{\mathrm{y}}_1+{\mathrm{m}}_2{\mathrm{y}}_2+{\mathrm{m}}_3{\mathrm{y}}_3}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3}$

      = $\frac{[1\times 0+{\displaystyle 2}\times 0+3\times {\displaystyle \frac{\sqrt{3}\mathrm{a}}{2}}]}{[1+{\displaystyle 3}+2]} = \frac{\sqrt{3}\mathrm{a}}{4}$