A genre studied in mathematics is similarity. Similarity is frequently given as a problem in figures. In similarity problems, there are proof problems and calculation problems.

Therefore, we need to learn the conditions under which figures are similar. Not only that, but you should be able to find the side lengths of similar figures. In this case, we will use the proportional expression.

Proportions are used frequently in our lives. Therefore, proportional calculations are important, and by thinking of calculation problems using similar figures in the same way as proportional ones, we can calculate the side lengths. You can also calculate the area ratio and volume ratio from the side lengths.

You can solve problems by finding similar figures and checking the ratio of the side lengths. Therefore, in order to understand the relationships of similarity between figures, we will explain the features of similar figures, similarity theorems, and similarity ratios.

Table of Contents

## What Is The Difference Between Congruence and Similarity: Properties of Similarity

In mathematics, we study congruence. There is a difference between congruence and similarity. Congruence refers to shapes that are exactly the same. On the other hand, figures that are the same in shape but different in size are called similarity.

For example, the following figures are in a similar relationship.

The shape of the figure is the same. However, the size is different. Understand that similarity is a shape in which the side lengths are enlarged or reduced. In addition, similar figures have the following properties

**-The length Ratio of the Corresponding Sides Are Equal**

Since the figures are enlarged or reduced, the length ratio of the corresponding sides is the same for each similar figure. For example, if one side is doubled in length, all the other sides are also doubled.

If the figures are similar and the side ratio is known, the other side lengths can be calculated.

**-The Sizes of the Corresponding Angles are Equal**

For the side lengths, all ratios are equal for similar figures, as mentioned above. On the other hand, for angles, they are the same for all pairs of angles. For example, we have the following.

If the angles are different, the shape of the figure will change. Therefore, it is not a similar shape. In the case of similarity, the angles are always the same, and the only difference is the side lengths.

**-The Sign Used in Similarity**

In addition, the symbol ∼ is used in similarity. For example, write △ABC∼△EDF. In this case, △ABC and △EDF are similar.

For congruence, we use the ≅ sign. In the same way, remember to use a special sign for similarity.

### Three Conditions for Triangles to be Similar

When do triangles become similar? There are three similarity theorems for triangles in total. Here are the similarity conditions for triangles.

- Side – Side – Side (SSS) Similarity Theorem
- Side – Angle – Side (SAS) Similarity Theorem
- Angle – Angle (AA) Similarity Theorem

The details of each are as follows.

**-Side – Side – Side (SSS) Similarity Theorem**

If the ratios of three pairs of sides are all equal, they are similar.

**-Side – Angle – Side (SAS) Similarity Theorem**

If the length ratios of two pairs of sides are equal and the angle between the sides is the same, the two figures are similar.

**-Angle – Angle (AA) Similarity Theorem**

If two pairs of angles are equal, then each shape is similar.

Note that, similarity of figures almost always uses Angle – Angle (AA). It is rare to have to prove similarity of triangles using Side – Side – Side (SSS) or Side – Angle – Side (SAS).

Therefore, in a similarity problem, the first step to consider is Angle – Angle (AA). Try to find triangles that are similar by looking for two sets of the same angles. After that, you can use the similarity theorem to prove that the triangles are similar.

## Relationship Between Similarity Ratios and Side Lengths

With similarity figures, it is not only important to prove that the figures are similar to each other. It is also important to use the similarity to calculate the side lengths. The calculation used in this case is the similarity ratio. What is the similarity ratio?

Similarity ratios should be understood as ratios of the side lengths. In a similar figure, the length ratios of the corresponding sides are the same. For example, in the following case, the similarity ratio is 1:3.

If the side length ratio is 1:3, the similarity ratio of the two triangles is 1:3.

**-The Similarity Ratio Can Use Height as Well as Side Length**

Depending on the similarity ratio, the side lengths will change. The same can be said for the height as well as the side length. For example, if the similarity ratio is 1:3, not only will all the side lengths be tripled, but the height will also be tripled, as shown below.

In similarity figures, every length is related to the similarity ratio.

### Calculating the Side Lengths Using the Proportional Expressions

Once you understand the similarity ratio, you can calculate the side lengths. The calculation method is the same as the proportional expression. For example, given the following similarity figure (△ABC∼△EDC), how long is the length of a?

The length of the BC is 3 cm and the length of the DC is 4 cm. Therefore, we know that the similarity ratio is 3:4.

By using this similarity ratio, let’s find the length of a. The length of AB is 6 cm. Therefore, by using the similarity ratio, we can create the following proportional equation.

The proportional expression can be calculated by outer multiplication and inner multiplication. In the previous proportional expression, the calculation can be done as follows.

After creating a proportional expression, be sure to remember the properties that can be calculated by outer multiplication and inner multiplication. Proportional equations are used in many aspects of mathematics, not just similarity.

For reference, we can see that the following proportional expressions all have the same value when multiplying outside and inside.

- $1:3=3:9$
- $2:5=6:15$
- $4:3=6:\displaystyle\frac{9}{2}$

When the proportions are the same, the numbers will always match when multiplying outside and inside. Using this property of proportional expressions, we can calculate the side lengths of similar figures.

### The Area Ratio Is Squared and the Volume Ratio Is Cubed

The side lengths can be calculated by using similarity ratios in this way. However, in similarity problems, you may be asked to calculate the area and volume. Therefore, we need to understand how the area ratio and volume ratio work.

In the case of area ratio, it is the square of the similarity ratio. For example, if the similarity ratio is 1:2, the area ratio is 1:4 (2^{2}). And if the similarity ratio is 1:3, the area ratio is 1:9 (3^{2}). Why is it that the square of the similarity ratio becomes the area ratio?

When calculating the area of a triangle, you can use the following formula.

- Triangle area = vertical × horizontal × $\displaystyle\frac{1}{2}$

As mentioned above, if the similarity ratio is doubled or tripled, the side lengths will be doubled or tripled. In area, this is calculated by multiplying the vertical and horizontal lengths. This is why the area ratio is the similarity ratio squared.

For example, if the similarity ratio is 1:2, the vertical and horizontal lengths are doubled, and the area ratio is quadrupled because two doubled sides are multiplied. In the case of a 1:3 similarity ratio, the vertical and horizontal lengths are tripled, and the area ratio is 9 times because two tripled sides are multiplied.

On the other hand, what about volume? The volume is calculated by the following formula.

- Volume = Length × Width × Height

In a similarity figure, if the length is doubled, the width and height are also doubled. Therefore, it becomes the cube of the similarity ratio.

For example, if the similarity ratio is 1:2, the length, width, and height are each doubled. Three doubled sides are multiplied, so the volume ratio is 8 times (2^{3} times). If the similarity ratio is 1:3, the length, width, and height are each tripled. Three tripled sides are multiplied, so the volume ratio is 27 times (3^{3} times).

## Exercise: Proof of Similarity and Calculation of Similarity Ratio

**Q1:** Solve the following problem.

There is a trapezoid ABCD with AD||BC, and the intersection point of AC and BD is O. AD is 3 cm, BC is 9 cm, and the area of △OAD is 12 cm^{2}.

- Prove that △OAD∼△OCB
- Calculate the area of △OCB
- Calculate the area of the trapezoid ABCD

**A1:** Answers.

**(a)**

- In △OAD and △OCB
- ∠AOD = ∠COB: Vertical Angles are equal – (1)
- ∠OAD = ∠OCB: Alternate angles with parallel lines are equal – (2)
- From (1) and (2), since Angle – Angle (AA), △OAD∼△OCB

**(b)**

Since AD = 3 cm and BC = 9 cm, the similarity ratio is 1:3. As mentioned above, the area ratio is the square of the similarity ratio. In other words, the area ratio is 1:9. Since the area of △OAD is 12cm^{2}, we can create the following proportional equation.

- $1:9=12:x$

Solving this proportional expression, we get the following.

- $x=9×12=108$

Therefore, the area of △OCB is 108 cm^{2}.

**(c)**

We have found the area of △OAD and △OCB. On the other hand, how can we calculate the area of △OAB and △ODC? If we know the areas of these triangles, we can add all the triangles together to get the area of the trapezoid.

It may seem difficult to find the area of △OAB and △ODC. On the other hand, it is possible to calculate the area of △ABC and △DBC. The similarity ratio between △OAD and △OCB is 1:3, so the height ratio between △ABC and △OCB is 4:3, as shown below. In the same way, the ratio of the heights of △DBC and △OCB is 4:3.

The base BC of the triangles are the same. The only difference is the height. Also, as the height increases, the area increases in proportion to the ratio. So by using the ratio of the heights, we can create the following proportional expression.

- $4:3=x:108$

Solving this proportional equation, we get the following.

- $3x=4×108$
- $x=144$

Therefore, the area of △ABC is 144cm^{2}. Subtracting the area of △OCB from this, we can get the area of △OAB. Therefore, the area of △OAB is 36cm^{2}.

- $△OAB=144-108=36$

The base and height lengths of △ABC and △DBC are the same. Also, by subtracting the area of △OCB, we can get the area. Therefore, the areas of △OAB and △ODC are the same.

Then, adding the areas of all the triangles, we find that the trapezoid area is 192 cm^{2}.

- $12+108+36+36=132$

## Using the Similarity Theorems to Solve Problems

Compared to the proof of congruence, the proof of similarity is easy: if you find that two pairs of angles are equal, then the two triangles are similar.

However, with similar figures, we rarely end up proving that they are similar. Furthermore, we need to solve advanced problems. By using the similarity ratio, we can calculate the side lengths.

In addition to the side lengths, you may also have to calculate the area and volume. In such cases, we need to use similarity ratios to find the area and volume ratios.

In similarity figures, you need to be able to solve advanced problems by using ratios. So let’s review how to calculate proportional expressions, and make sure that you can even calculate area and volume.