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Area and Perimeter of Similar Triangles

What is true about the ratio of the area of similar triangles?

Answer: The ratio of the areas is the square of the similarity ratio (aka the scale factor).
It's easiest to see that this is true if you look at some specific examples of real similar triangles.

$$ A= \frac{1}{2}\cdot{12}\cdot{4} \\ =24 $$

$$ A= \frac{1}{2}\cdot{24}\cdot{8} \\ =96 $$

Notice: $$ \frac{24}{96} = \frac{1}{4} $$

Therefore, if you know the similarity ratio, all that you have to do is square it to determine ratio of the triangle's areas.

What about the perimeter of similar triangles?

Answer: The ratio of the perimeters is equal to the similarity ratio (aka the scale factor).
Let's look at the two similar triangles below to see this rule in action.
Perimeter of Triangle #1
question
Perimeter of Triangle #2
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Conclusion
Practice Problems


Problem 1)
$$\triangle ABC$$ ~ $$\triangle XYZ$$ and have a scale factor (or similarity ratio) of $$ \frac{3}{2} $$.
What is the ratio of their areas?
Answer


Problem 2)
$$\triangle ABC$$ ~ $$\triangle XYZ$$. The ratio of their perimeters is $$ \frac{11}{5} $$, what is their similarity ratio and the ratio of their areas?
What is the ratio of their areas?
Answer


Problem 3)
$$\triangle ABC$$ ~ $$\triangle XYZ$$. The ratio of their areas is $$ \frac{36}{17} $$, what is their similarity ratio and the ratio of their perimeters?
What is the ratio of their areas?
Answer


Problem 4)
$$\triangle HIJ$$ ~ $$\triangle XYZ$$. The ratio of their areas is $$ \frac{25}{16}$$, if XY has a length of 40, what is the length of HI?
Answer