Sign In
If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
Let KLMN and PQRS be similar figures, and A_1 and A_2 be their respective areas. The length scale factor between corresponding side lengths is ab. Here, the following conditional statement holds true.
KLMN ~ PQRS ⇒ A_1/A_2 = (a/b )^2
The area of a rectangle is the product of its length and its width.
| Area of KLMN | Area of PQRS |
|---|---|
| A_1 = KL* LM | A_2 = PQ * QR |
By the definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor ab. KL/PQ= a/b [1.1em] LM/QR=a/b ⇔ KL = PQ * a/b [1.1em] LM = QR * a/b The next step is to substitute the expressions for KL and LM into the formula for A_1, which represents the area of KLMN.
KL= PQ * a/b, LM= QR * a/b
Remove parentheses
Commutative Property of Multiplication
a* a=a^2
Associative Property of Multiplication
Notice that the expression on the right-hand side is ( ab )^2 times the area of PQRS, or A_2.
PQ* QR= A_2
.LHS /A_2.=.RHS /A_2.
This proof has shown that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths. This ratio is also called the area scale factor.
Scale Factor & & Area Scale Factor a/b & ⇒ & A_1/A_2 = (a/b )^2