**Solution:**

Let the minimum length of the shortest side be x.

Then the longest side of a triangle is 3x and the third side is 3x – 2.

Perimeter of the triangle ≥ 61

x + 3x + 3x – 2 ≥ 61

7x – 2 ≥ 61

7x ≥ 61 + 2

7x ≥ 63

x ≥ 9

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