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Rule

Perimeters of Similar Polygons Theorem

If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths.

Let P_1 and P_2 be the perimeters of QRST and ABCD, respectively. Let ab be the scale factor between corresponding side lengths. Then, based on the above diagram, the following relation holds true.

ABCD ~ QRST ⇒ P_1/P_2 =a/b

Proof

Let QRST and ABCD be two similar polygons with perimeters P_1 and P_2, respectively. By the definition of similar polygons, corresponding side lengths are proportional, with the scale factor ab as the common ratio. QR/AB=a/b [0.8em] RS/BC=a/b [0.8em] ST/CD=a/b [0.8em] TQ/DA=a/b ⇔ QR = a/b * AB [0.8em] RS = a/b * BC [0.8em] ST = a/b * CD [0.8em] TQ = a/b * DA If the equations are added together, the left-hand side of the resulting equation will give the perimeter of QRST. The right-hand side will give the scale factor times the perimeter of ABCD.

QR+RS+ST+TQ=a/b * AB+a/b * BC+a/b * CD+a/b * DA

FactorOut

Factor out a/b

QR+RS+ST+TQ=a/b (AB+BC+CD+DA)

SubstituteII

QR+RS+ST+TQ= P_1, AB+BC+CD+DA= P_2

P_1=a/b * P_2

Finally, the Perimeters of Similar Polygons Theorem is obtained by dividing both sides by P_2.

P_1=a/b * P_2 ⇔ P_1/P_2=a/b

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