The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side.

Let the length of the shortest side of the triangle be $x \mathrm{~cm}$.

Then, length of the longest side $=3 x \mathrm{~cm}$

Length of the third side $=(3 x-2) \mathrm{cm}$

Since the perimeter of the triangle is at least $61 \mathrm{~cm}$,

$x \mathrm{~cm}+3 x \mathrm{~cm}+(3 x-2) \mathrm{cm} \geq 61 \mathrm{~cm}$

$\Rightarrow 7 x-2 \geq 61$

$\Rightarrow 7 x \geq 61+2$

$\Rightarrow 7 x \geq 63$

$\Rightarrow \frac{7 x}{7} \geq \frac{63}{7}$

$\Rightarrow x \geq 9$

Thus, the minimum length of the shortest side is 9 cm.