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
Perimeter $$ = 6 + 8 + 10 = 24 $$

Perimeter of Triangle #2
Perimeter $$ = 5 + 3 +4 = 12 $$

The ratio of the perimeter's is exactly the same as the similarity ratio!
$
\frac{ \text{perimeter #1}}{ \text{perimeter #2}} = \frac{24}{12}= \frac{2}{1}
$
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?
$
\text{ratio of areas} = (\text{similarity ratio})^2
\\
= \Big( \frac{3}{2}\Big )^2
\\
= \frac{9}{4}
$
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?
$
\text{ratio of perimeters} = \text{similarity ratio}
\\
\text{similarity ratio} = \frac{11}{5}
\\
\text{ratio of areas} = (\text{similarity ratio})^2
\\
= \Big( \frac{11}{5}\Big )^2
\\
\text{ratio of areas} = \frac{121}{25}
$
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?
$
\text{ratio of areas} = (\text{similarity ratio})^2
\\
(\text{similarity ratio})^2 = \text{ratio of areas}
\\
\text{similarity ratio} = \sqrt{ \text{ratio of areas} }
\\
= \sqrt{ \Big( \frac{36}{17} \Big) }
\\
= \frac{ \sqrt{36}}{ \sqrt{ 17 } }
\\
= \frac { 6 }{ \sqrt{ 17 } }
$
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?
We need to find the similarity ratio first, since that ratio gives us a proportion between corresponding sides.
$
\text{ratio of areas} = (\text{similarity ratio})^2
\\
(\text{similarity ratio})^2 = \text{ratio of areas}
\\
\text{similarity ratio} = \sqrt{ \text{ratio of areas} }
\\
= \sqrt{ \Big( \frac{25}{16} \Big) }
\\
\text{similarity ratio} = \frac{5}{ 4 }
$
Now, that you have found the similarity ratio, you can set up a proportion to solve for HI
$
\frac{5}{ 4 } = \frac{HI}{XY}
\\
\frac{5}{ 4 } = \frac{HI}{40}
\\
\frac{40 \cdot 5}{ 4 } = HI
\\
HI = 50
$
