
How could you model the positions of the particles mathematically?

Why not assign the positions of the particles and the centre of the disc using Cartesian coordinates?

How about assigning a fixed position for one particle, then finding the positions of the other two?

Assuming the centre of the disc is at
$(0,0)$
. For the disc to balance at its centre, the particles must satisfy the centre of mass equation$\sum w_i\mathbf{x}_i = \mathbf{0},$
where$w_i, \mathbf{x}_i$
denote the weight and the vector position of particle$i$
respectively.

The centre of mass must be at the centre of the disc, or else it tips. Without loss of generality, place the centre of the disc at
$(0,0),$
the 3 kg point at$(0,1)$
and the two 2 kg points at$(x,y)$
and$(x,y)$
respectively. Writing the centre of mass equation on the$y$
axis, we get$0 = {3+4y\over 3},$
which gives$y={3\over4}.$
Using Pythagoras’ Theorem, we then have$x={\sqrt{7}\over4}.$
The total area of the triangle is thus$\frac{2x(1+y)}{2}={7\sqrt{7}\over16}.$