Similar figures are characterized by always having the same shape, but not the same size. These figures have equal angles, but not equal sides. The proportions of the corresponding sides of similar figures is called the scale factor and can be used to describe how big or how small one figure is compared to another.

Here, we will learn about similar figures and their scale factors in more detail.

## Definition of similar figures

Two or more figures are considered similar when they have the same shape. Similar figures can have the same size or one can be larger than the other.

We use the symbol “~” to indicate that the two figures are similar. For example, if the triangles ABC and DEF are similar, we can write:

ΔABC ~ ΔDEF

Since the shape of similar figures is the same, their corresponding angles will also be the same. We can use the similarity statement to determine which angles are equal. For example, in the example above, angles A and D are the same since they are written in first order.

Similarly, angles B and E are also equal, as are angles C and F.

Similar figures may have sides that have different sizes, so the corresponding lengths are not necessarily the same. However, the proportions of the corresponding angles are always the same. For example, in the following similar figures, we have the proportions:

Now, if we divide all the corresponding sides of the following similar figures, we always get 1/2. For two figures to be similar, the proportions of their corresponding sides must be the same.

## What are scale factors?

The scale factors are the proportions that we obtain when we divide the lengths of the corresponding sides of similar figures. In the example we saw above, all the proportions simplify to 1/2, so we have that the scale factor from triangle ABC to triangle DEF is 1/2.

We can also consider the scale factors as multipliers. For example, the scale factor from triangle MNO to triangle PQR is 2. This means that triangle PQR is twice as large as triangle MNO.

If we multiply one side of triangle MNO by 2, we get the length of the corresponding side of triangle PQR. The proportions of all the sides of these triangles are equal to 2. That is, we have $latex \frac{10}{5}=2$, $latex \frac{12}{6}=2$, $latex \frac{6}{3}=2$.

Let’s look at another example. The scale factor from rectangle ABCD to rectangle MNOP is 3. This means that the second rectangle is three times as large compared to the first rectangle. We can get the lengths of the corresponding sides of the second rectangle by multiplying the sides of the first rectangle by 3.

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## Examples of scale factors

### EXAMPLE 1

- Find the scale factor from the trapezoid ABCD to the trapezoid EFGH.

**Solution:** In this case, the small figure was indicated first. Since the second figure is larger, this means that the scale factor must be greater than 1. Now, comparing the lengths of the corresponding sides, we can deduce that we can multiply by 2 to obtain the same lengths.

Therefore, the scale factor is 2. We can also find the scale factor by finding the following ratios:

$latex \frac{6}{3}=2, ~\frac{10}{5}=2, ~\frac{18}{9}=2$

### EXAMPLE 2

- Find the scale factor from triangle MNO to triangle PQR.

**Solution:** The second triangle indicated is larger than the first, so we know that the scale factor is going to be greater than 1. In this case, the scale factor is not a whole number, so we cannot determine it at first sight. Therefore, we can find the proportions of the corresponding sides.

We know that the scale factor is greater than 1, so we put the larger number in the numerator of the fractions. Therefore, we have the ratios:

$latex \frac{8}{4}=2,~\frac{10}{5}=2, ~\frac{6}{3}=2$

Therefore, the scale factor is 2.

### EXAMPLE 3

- Find the scale factor from triangle PQR to triangle MNO.

**Solution:** This example is similar to the previous example with the difference that the order is swapped. Since the second triangle is smaller, the scale factor will be less than 1 in this case.

We know that the order has just been changed, so we can flip the fractions to find the correct ratios:

$latex \frac{4}{8}=\frac{1}{2}, ~\frac{3}{6}=\frac{1}{2}, \frac{5}{10}=\frac{1}{2}$

Therefore, the scale factor is $latex \frac{1}{2}$.

## See also

Interested in learning more about similar figures, congruency, and other geometry topics? Take a look at these pages:

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