Question

prove that if two triangles are similar then the ratio of their perimeters is equal to the ratio of the corresponding sides

Answer 1

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

Proved.