# Similarity ratio

🏆Practice the ratio of similarity

## What is the similarity ratio?

The similarity ratio is the constant difference between the corresponding sides of the two shapes.
That is, if the similarity ratio is $3$, we know that each side of the large triangle is $3$ times larger than that of the small triangle.

## How do we calculate the similarity ratio?

The calculation of the similarity ratio is divided into several steps that must be performed:

1. First we must know that we are dealing with similar triangles or polygons.
2. We must know how to identify the corresponding sides in each of the triangles or polygons.
3. We need to know the sizes of a pair of equal sides.
4. We must divide the size of one side by the size of the other side.

The result obtained is actually the similarity ratio.

## Test yourself on the ratio of similarity!

Is the similarity ratio between the three triangles equal to one?

We will exemplify this through an exercise.

In the drawing before us there are two triangles similar to $ABC$ and $KLM$.

We are required to calculate the similarity ratio between the two triangles.

We are going to work according to the steps described above.

The first step is actually completed - It has been given to us, because these are two similar triangles.

In the second step, we must identify the corresponding sides in each of the two triangles. We will look at the drawing and see that the two triangles have angles. The angle $A$ is equal and equal to the angle $K$ and the angle $B$ is equal to the angle $L$.

From this we can conclude that, in terms of location, the sides $AB$ and $KL$ are corresponding sides.

The third step is fairly easy, because we are given the sizes of these two sides, $AB=3$, $KL=6$

In the fourth and final step, we will perform a simple operation of dividing the sizes of the corresponding sides.

We obtain:

${KL \over AB} = {6 \over 3} = 2$

We obtain that the similarity ratio of these two similar triangles is equal to $2$.

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## Similarity ratio exercises

### Exercise 1

Given:

$ΔACB∼ΔBED$

Solution

According to $A.A$ the two triangles are similar.

Answers $a + b$

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### Exercise 2

Similar triangles:

$\frac{BC}{EF}=\text{?}$

Solution

The triangles are similar as a result of the similarity ratio.

$\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}$

$\frac{10}{5}=2$

$2$

### Exercise 3

Question

What is the ratio between the sides of the triangles $ΔABC$ and $ΔAMN$?

Solution

From image comes out:

$\frac{BC}{MN}=\frac{6}{3}=2$

$\frac{BC}{MN}=2$

Exercise 4:

Solution

Ratio of similarity

$\frac{AD}{AF}=\frac{1}{4}$

$\frac{AF}{AB}=\frac{1}{4}$

$\frac{AD}{AF}=\frac{AF}{AB}$

Do you know what the answer is?

### Exercise 5

Question

Find the similarity ratio corresponding to the triangles $ΔDEF$ and $ΔABC$.

Solution

$\sphericalangle C=\sphericalangle F=54=\frac{AB}{ED}$

$\sphericalangle B=\sphericalangle E=60=\frac{AC}{FD}$

$\sphericalangle A=\sphericalangle D=66=\frac{BC}{FE}$

It follows that

$\frac{25}{75}=\frac{1}{3}$

$\frac{BC}{FE}=\frac{AC}{FD}=\frac{AB}{DE}=\frac{1}{3}$

$\frac{1}{3}$

## Review questions

What is the similarity ratio?

It is the ratio of the corresponding sides of two similar figures.

How to get the similarity ratio?

The similarity ratio is obtained by dividing the corresponding sides of two similar figures. Let's see an example:

Given the following similar triangles. $\triangle ABC\sim\triangle DEF$

Calculate the similarity ratio

Given that $\triangle ABC\sim\triangle DEF$

Then we must locate which are the corresponding sides, and from this we deduce that

$\sphericalangle A=\sphericalangle D$

$\sphericalangle B=\sphericalangle E$

Then the corresponding sides are $AB$,$DE$

Now to calculate the similarity ratio we do the quotient of these two sides.

$\frac{AB}{DE}=\frac{12}{10}=\frac{6}{5}=1.2$

Therefore the similarity ratio is $1.2$

What are two similar triangles?

We can say that two triangles are similar when they have the same shape even if they have different sizes, for that they must meet some of the following similarity criteria:

• Side-Side-Side (SSS): If the ratio of their three pairs of corresponding sides is the same then two triangles are similar.
• Side-Angle-Side (SAS): Two triangles are similar if the ratio of two pairs of corresponding sides is the same and the angle between these two pairs is the same, then they are similar triangles.
• Angle-Angle (AA): For two triangles to be similar by this criterion, two of their respective angles must measure the same and therefore the third angle must also have the same measure as the angle corresponding to that angle. That is, their three corresponding angles measure the same.

What are congruent figures?

Unlike similar figures that do not necessarily have to equal side lengths, two figures are congruent when they have the same shape AND their corresponding sides are equal lengths.

What is the similarity ratio of two rectangles?

Just like similar triangles, to calculate the similarity ratio we must calculate the quotient of the corresponding sides. Let's see an example:

Given the following similar rectangles

$ABCD\sim EFGH$

Find the similarity ratio

Since they are similar rectangles and because they are quadrilaterals they have right angles, then we can deduce their corresponding sides:

One of their corresponding sides are $AD$,$EH$, then we can calculate the similarity ratio.

$\frac{EH}{AD}=\frac{10}{4}=\frac{5}{2}=2.5$

Therefore the similarity ratio is $2.5$

## Examples with solutions for The Ratio of Similarity

### Exercise #1

Is the similarity ratio between the three triangles equal to one?

### Step-by-Step Solution

To answer the question, we first need to understand what "similarity ratio" means.

In similar triangles, the ratio between the sides is constant.

In the statement, we do not have data on any of the sides.

However, a similarity ratio of 1 means that the sides are exactly the same size.

That is, the triangles are not only similar but also congruent.

In the drawing, you can clearly see that the triangles are of different sizes and, therefore, clearly the similarity ratio between them is not 1.

No

### Exercise #2

According to which theorem are the triangles similar?

What is their ratio of similarity?

### Step-by-Step Solution

According to the given data, we will write the side ratios as follows:

$\frac{FD}{AB}=\frac{X}{2X}=\frac{1}{2}$

$\frac{FE}{AC}=\frac{\frac{y}{2}}{y}=\frac{y}{2y}=\frac{1}{2}$

$\frac{DE}{BC}=\frac{2Z}{4Z}=\frac{2}{4}=\frac{1}{2}$

From this, it follows that the ratio is according to S.S.S (Side-Side-Side):

$\frac{FD}{AB}=\frac{FE}{AC}=\frac{DE}{BC}=\frac{1}{2}$

S.S.S., $\frac{1}{2}$

### Exercise #3

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.

### Step-by-Step Solution

Let's look at the order of letters of the triangles that match each other and see the ratio of the sides.

We will write accordingly:

Triangle ABC is similar to triangle DFE

The order of similarity ratio will be:

$\frac{AB}{DF}=\frac{BC}{FE}=\frac{AC}{DE}$

Now let's insert the existing data we have in the diagram:

$\frac{8y}{9y}=\frac{7m}{FE}$

Let's reduce y and we get:

$\frac{8}{9}FE=7m$

$FE=\frac{9}{8}\times7m$

$FE=7\frac{7}{8}m$

$7\frac{7}{8}m$

### Exercise #4

$ΔACB∼ΔBED$

### Step-by-Step Solution

First, let's look at angles C and E, which are equal to 30 degrees.

Angle C is opposite side AB and angle E is opposite side BD.

$\frac{AB}{DB}$

Now let's look at angle B, which is equal to 90 degrees in both triangles.

In triangle ABC the opposite side is AC and in triangle EBD the opposite side is ED.

$\frac{AC}{ED}$

Let's look at angles A and D, which are equal to 60 degrees.

Angle A is the opposite side of CB, angle D is the opposite side of EB

$\frac{CB}{EB}$

Therefore, from this it can be deduced that:

$\frac{AB}{BD}=\frac{AC}{ED}$

And also:

$\frac{CB}{ED}=\frac{AB}{BD}$

Answers a + b are correct.

### Exercise #5

What is the ratio between the sides of the triangles ΔABC and ΔMNA?

### Step-by-Step Solution

From the data in the drawing, it seems that angle M is equal to angle B

Also, angle A is an angle shared by both triangles ABC and AMN

That is, triangles ABC and AMN are similar respectively according to the angle-angle theorem.

According to the letters, the sides that are equal to each other are:

$\frac{AB}{AM}=\frac{BC}{MN}=\frac{AC}{AN}$

Now we can calculate the ratio between the sides of the given triangles:

$MN=3,BC=6$$\frac{6}{3}=2$

$\frac{BC}{MN}=2$