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Here are a few recommended readings before getting started with this lesson.
Two polygons are said to be similar if their corresponding sides are proportional and their corresponding angles are congruent. Because of this, there is a relation between the perimeters of similar polygons.
If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths.
Let P1 and P2 be the perimeters of QRST and ABCD, respectively. Let ba be the scale factor between corresponding side lengths. Then, based on the above diagram, the following relation holds true.
ABCD∼QRST⇒P2P1=ba
Factor out ba
QR+RS+ST+TQ=P1, AB+BC+CD+DA=P2
P1=ba⋅P2⇔P2P1=ba
Tadeo likes playing basketball. He decides to make a miniature model of a basketball court with a length of 32 centimeters.
A standard basketball court has a length of 28 meters and a width of 15 meters.
Substitute values
ba=b/16a/16
P2=86, Scale factor=1752
LHS⋅86=RHS⋅86
ca⋅b=ca⋅b
Calculate quotient
Round to 2 decimal place(s)
Like with perimeters, there is a relation between the areas of similar polygons.
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 A1 and A2 be their respective areas. The length scale factor between corresponding side lengths is ba. Here, the following conditional statement holds true.
KLMN∼PQRS⇒A2A1=(ba)2
The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.
The area of a rectangle is the product of its length and its width.
Area of KLMN | Area of PQRS |
---|---|
A1=KL⋅LM | A2=PQ⋅QR |
KL=PQ⋅ba, LM=QR⋅ba
Remove parentheses
Commutative Property of Multiplication
a⋅a=a2
Associative Property of Multiplication
Substitute values
ba=b/12a/12
y=3, z=5
Calculate power
LHS−9=RHS−9
LHS=RHS
a2=a