Let the length of shortest side = ‘x’ cm

According to the question,

The longest side of a triangle is twice the shortest side

⇒ Length of largest side = 2x

Also, the third side is 2 cm longer than the shortest side

⇒ Length of third side = \[\left( x\text{ }+\text{ }2 \right)\text{ }cm\]

Perimeter of triangle = sum of three sides

\[\begin{array}{*{35}{l}}

=\text{ }x\text{ }+\text{ }2x\text{ }+\text{ }x\text{ }+\text{ }2  \\

=\text{ }4x\text{ }+\text{ }2\text{ }cm  \\

\end{array}\]

Now, we know that,

Perimeter is more than 166 cm

\[\begin{array}{*{35}{l}}

\Rightarrow ~4x\text{ }+\text{ }2\text{ }\ge \text{ }166  \\

\Rightarrow ~4x\text{ }\ge \text{ }164  \\

\Rightarrow ~x\text{ }\ge \text{ }41  \\

\end{array}\]

Hence, minimum length of the shortest side should be = 41 cm.