1. Assign coordinates to the three identical masses

Since the triangle is right-angled with mutually perpendicular sides of length $3\,\text{m}$, and the point of intersection of these perpendicular sides is taken as the origin, the three corner coordinates are:

• Origin: $(0,0)$
• Along x-axis: $(3,0)$
• Along y-axis: $(0,3)$

Each sphere has mass $M$.

2. Use the centre of mass formula

For equal masses, the coordinates of the centre of mass are simply the averages of the coordinates:

$x_{\text{CM}} = \frac{0+3+0}{3} = 1$

$y_{\text{CM}} = \frac{0+0+3}{3} = 1$

So, the position vector of the centre of mass is

$\vec r_{\text{CM}} = \hat i + \hat j$

3. Find the magnitude of the position vector

$|\vec r_{\text{CM}}| = \sqrt{1^2 + 1^2} = \sqrt{2}\,\text{m}$

Given that this magnitude is $\sqrt{x}\,\text{m}$, we have

$\sqrt{x} = \sqrt{2}$

Therefore,

$x = 2$

4. Compare with stored answer

Stored correct answer = $2$

Our derived answer = $2$

So they agree.