Ratio of areas of similar triangles
To find the ratio of the areas of similar triangles, just square the similarity ratio.
The ratio of the perimeters on the other hand equals the similarity ratio.
Why area's ratio is the similarity's squared The similarity ratio from triangle #1 to #2 is ½. We can use the formula for the area of a triangle to find that Area of Triangle #1 = ½(12 • 4) = 24
Area of Triangle #2 = ½(24 • 8) = 96
As you can see from our knowledge of the formula of area, the ratio of the areas is
When two triangles are similar, the reduced ratio of any two corresponding sides is called the
scale factor of the similar triangles. In Figure
1 , Δ
ABC∼ Δ
DEF.
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| Figure 1 | Similar triangles whose scale factor is 2 : 1. |
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Theorem 60: If two similar triangles have a scale factor of
a :
b, then the ratio of their perimeters is
a :
b.
Theorem 61: If two similar triangles have a scale factor of
a :
b, then the ratio of their areas is
a2 :
b2.
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