### Introduction to Similar Figures

Similar figures are two figures (or shapes) that have the same angles and proportional dimensions. In this lesson we look at how to find the similarity ratio (or similitude), and using that to determine the lengths of the side a of similar polygons, and how to find the height of a tree from its shadow.

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## What are Similar Figures?

What does it mean when we say we have 2 similar figures? Similar figures can be thought of as a figure and an enlargement or reduction of that figure, without any distortions. Scale models are one way that the concepts of similar figures is used.

For two polygons to be considered similar, their corresponding angles must have the same measurement and their corresponding sides need to be proportional.

The ratio of the lengths of corresponding sides in two similar figures is the similarity ratio (or similitude).

Let’s start by looking at the parallelograms CATS and BIRD. We can tell they are similar because all the angles are equal and the similarity ratios for the corresponding sides are equal.

$\dfrac{CS}{BD}=\dfrac{2.1}{3.15} = \dfrac{2}{3}$ * (or $0.\overline{6}$) *and $\dfrac{ST}{DR}=\dfrac{3}{4.5}= \dfrac{2}{3}$ *(or $0.\overline{6}$)*

## Similar Figures: How is it useful?

The parallelogram FOXZ is similar to CATS and BIRD.

So, what do we know about $m\angle F$?

Well, $\angle F$ corresponds to $\angle B$, so the measurement must be $110^o$.

Likewise, $\angle Z$ corresponds to $\angle D$, so the measurement must be $70^o$.

How can we find the value of $\overline{OX}$?

The similarity ratio for $\frac{CATZ}{FOXZ}=\frac{3}{5}$, since $\overline{ST}$ and $\overline{OX}$ are corresponding parts. $\overline{OX}=n$ is located on the polygon FOXZ, so we would put it across from the $5$ in our ratio, and the corresponding side on CATS is $\overline{AT}=2.1$ so we will put it across from the $3$.

$\dfrac{CATS}{FOXZ}=\dfrac{3}{5} =\dfrac{2.1}{n}$

Now we can solve for the length of $n$ by cross multiplying.

$3n = 5\ast2.1$

$3n = 10.5$

$n = 3.5$

How to find the height of a tree, using similar figures.

Well, we are going to be looking at the ratio of the yardstick to the tree.

That ratio is equal to the yard stick’s shadow, which is 5 ft., to the tree’s shadow, which is 35 ft.

$\dfrac{yardstick}{tree}=\dfrac{5}{35} =\dfrac{1}{7}$

And that would be equal to the ratio of the height of the yard stick to the height of the tree, which is what we are trying to find out.

$\dfrac{yardstick}{tree}=\dfrac{1}{7} =\dfrac{3}{x}$

We can now use equivalent fractions to find the height of the tree.

$\dfrac{yardstick}{tree}=\dfrac{1\ast 3}{7 \ast 3} =\dfrac{3}{\color{red}{21}}$

So, the tree is 21 ft tall.