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Let ΔABC and ΔPQR are similar.

So \frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR}
Let the ratio of sides = \frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR} =x

We need to show that the ratio of perimeters is also x.

perimeter of ΔABC = AB+BC+AC
perimeter of ΔPQR = PQ+QR+PR

\frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR} =x\\\\ \Rightarrow AB=x.PQ\\AC=x.PR\\BC=x.QR

So perimeter of ΔABC = x.PQ + x.PR + x.QR = x(PQ+PR+QR)

ratio of perimeters =  \frac{Perimeter\ of \Delta ABC}{Perimeter\ of \Delta PQR}

⇒ Ratio =  \frac{AB+BC+AC}{PQ+QR+PR} = \frac{x(PQ+QR+PR)}{PQ+QR+PR} =x


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