# 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 $$= 6 + 8 + 10 = 24$$

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

$\text{ratio of areas} = (\text{similarity ratio})^2 \\ = \Big( \frac{3}{2}\Big )^2 \\ = \frac{9}{4}$

$\text{ratio of perimeters} = \text{similarity ratio} \\ \text{similarity ratio} = \frac{11}{5} \\ \text{ratio of areas} = (\text{similarity ratio})^2 \\ = \Big( \frac{11}{5}\Big )^2 \\ \text{ratio of areas} = \frac{121}{25}$

$\text{ratio of areas} = (\text{similarity ratio})^2 \\ (\text{similarity ratio})^2 = \text{ratio of areas} \\ \text{similarity ratio} = \sqrt{ \text{ratio of areas} } \\ = \sqrt{ \Big( \frac{36}{17} \Big) } \\ = \frac{ \sqrt{36}}{ \sqrt{ 17 } } \\ = \frac { 6 }{ \sqrt{ 17 } }$

We need to find the similarity ratio first, since that ratio gives us a proportion between corresponding sides.

$\text{ratio of areas} = (\text{similarity ratio})^2 \\ (\text{similarity ratio})^2 = \text{ratio of areas} \\ \text{similarity ratio} = \sqrt{ \text{ratio of areas} } \\ = \sqrt{ \Big( \frac{25}{16} \Big) } \\ \text{similarity ratio} = \frac{5}{ 4 }$

$\frac{5}{ 4 } = \frac{HI}{XY} \\ \frac{5}{ 4 } = \frac{HI}{40} \\ \frac{40 \cdot 5}{ 4 } = HI \\ HI = 50$