Let $ABCD$ and $PQRS$ be two similar polygons with perimeters $P_1$ and $P_2$ respectively. By definition of similar polygons, the side lengths are proportional and equal to the scale factor $k.$ $\begin{gathered} k = \dfrac{PQ}{AB} = \dfrac{QR}{BC} = \dfrac{RS}{CD} = \dfrac{SP}{DA} \end{gathered}$ From the proportions, the following four equations can be set. $\begin{cases} PQ = k\cdot AB \\ QR = k\cdot BC \\ RS = k\cdot CD \\ SP = k\cdot DA \end{cases}$ By adding these equations, the left-hand side will become equal to $P_1,$ while the right-hand side will become $k\cdot P_2.$ $\begin{aligned} PQ &= k\cdot AB \\ QR &= k\cdot BC \\ {\large+}\quad RS &= k\cdot CD \\ SP &= k\cdot DA \\\hline P_1 &= k\cdot P_2 \end{aligned}$ Finally, by dividing both sides by $P_2$ and substituting $k=\frac{PQ}{AB},$ the required result is obtained.