How Do You Find the Ratio of Similarity?

A ratio tells us something about the relationship between two objects. When you work with ratios, you have to know which sides belong together between different shapes. These are called corresponding sides.

Ratios are closely connected to similarity, because you use ratios when you check if two shapes are similar, or when you work with similarity in general.

Theory

Corresponding Sides

Corresponding sides are the sides that represent the same side in similar shapes. The symbol for corresponding sides is $≃$ . The ratio of two corresponding sides is called the linear ratio number $n$ .

Example 1

You are told that the sides in an arbitrary triangle $△ A B C$ have the lengths $A B = 2$ , $B C = 4$ and $C A = 5$ . Triangle $△ D E F$ has side lengths $D E = 4$ , $E F = 8$ and $F D = 9$ . Is triangle $△ A B C$ similar to $△ D E F$ ?

To keep the sides in order it is smart to draw a figure.

Figure:

Ratios of two triangles

Now you calculate the ratio between all the corresponding sides. If all the ratios are the same, the triangles are similar:

\begin{array}{l} \frac{A B}{D E} & = \frac{2}{4} = 0.5 \\ \frac{B C}{E F} & = \frac{4}{8} = 0.5 \\ \frac{C A}{D F} & = \frac{5}{9} = 0.56 \end{array}

You can see that one of the ratios is different, so you know that the triangles are not similar.

Theory

Ratio of Lengths

The ratio of the lengths of two corresponding sides $a$ and $b$ is the linear ratio number

n = \frac{a}{b}

Note! Whatever comes first in the problem goes in the numerator!

Example 2

Find the ratio of 12 to 24

Because 12 comes first in the problem, it goes in the numerator. Then the calculation is as follows:

ratio = \frac{12}{24} = \frac{1}{2} = 0.5

Theory

Ratio of Areas

The ratio of the areas of two similar shapes is the linear ratio number squared,

n^{2}

that is to say

{(\frac{a}{b})}^{2}

Example 3

Find the ratio of the areas of the similar triangles $△ A B C$ and $△ D E F$ . Triangle $△ A B C$ has $A B = 4$ , $B C = 5$ and $C A = 3$ . Triangle $△ D E F$ has $D E = 8$ , $E F = 10$ and $F D = 6$ .

Ratios of two right triangle

There are two ways to calculate the ratio of the areas. You can use the easy method described in the box above, or the hard method where you first calculate the areas, then calculate the ratio. You have to know both.

Method 1

Find the ratio by dividing two corresponding sides. You can choose the pair of sides you want:

n = \frac{A B}{D E} = \frac{4}{8} = \frac{1}{2}

Now you square $n$ and find the ratio of the areas:

\begin{array}{l} ratio of the areas & = n^{2} \\ = {(\frac{1}{2})}^{2} \\ = \frac{1}{4} \\ = 0.25 \end{array}

Method 2

First, you calculate the area of each triangle. If you have a keen eye, you will see that the triangles are right triangles, so you can use the formula for the area of a triangle directly. If you don’t see this, you first have to decide if the triangles are right triangles, or if you have to draw a help line and do multiple calculations to find the area.

Now that you know that they are right triangles, you can calculate the area directly.

\begin{array}{l} A_{△ A B C} & = \frac{4 \cdot 3}{2} = \frac{12}{2} = 6 \\ A_{△ D E F} & = \frac{8 \cdot 6}{2} = \frac{48}{2} = 24 \end{array}

Now you find the ratio of the answers:

Ratio = \frac{6}{24} = \frac{1}{4} = 0.25

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