Start with a rectangle that has a perimeter of 210 mm and one side is one-sixth the length of the other.

Let the length of one side of the rectangle is x, and the other is y. Then, according to the condition of the problem, we get:

1.2 * x + 2 * y = 210

2.6 * x = y

Substitute the resulting equation 2 into equation 1 and solve it:

2 * x + 2 * 6 * x = 210

2 * x + 12 * x = 210

14 * x = 210

x = 15

Substitute this value into Equation 2:

y = 6 * 15 = 90

Answer: The rectangle has sides of 15 and 90 mm.