# The Ratio of Similarity - Examples, Exercises and Solutions

## 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.

## 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):

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

### Answer

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:

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

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

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

### Answer

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

### Exercise #4

$ΔACB∼ΔBED$

Choose the correct answer.

### 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:

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

### Answer

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$

### Answer

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

### Exercise #6

In the image there are a pair of similar triangles and a triangle that is not similar to the others.

Determine which are similar and calculate their similarity ratio.

### Step-by-Step Solution

Triangle a and triangle b are similar according to the S.S.S (side side side) theorem

And the relationship between the sides is identical:

$\frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}$

$\frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3$

That is, the ratio between them is 1:3.

### Answer

$a$ and $b$, similarity ratio of $3$

### Exercise #7

The triangles above are similar.

Calculate the perimeter of the larger triangle.

### Step-by-Step Solution

We calculate the perimeter of the smaller triangle (top):

$3.5+1.5+4=9$

Due to their similarity, the ratio between the sides of the triangles is equal to the ratio between the perimeters of the triangles.

We will identify the perimeter of the large triangle using $x$:

$\frac{x}{9}=\frac{14}{3.5}$

$3.5x=14\times9$

$3.5x=126$

$x=36$

36

### Exercise #8

The similarity ratio between two similar triangles is 7, so that the area ratio is $_{——}$

### Step-by-Step Solution

We square it. 7 squared is equal to 49.

49

### Exercise #9

The triangle above are similar.

What is the perimeter of the blue triangle?

### Step-by-Step Solution

The perimeter of the left triangle: 13+12+5=25+5=30

Therefore, the perimeter of the right triangle divided by 30 is equal to 5.2 divided by 13:

$\frac{x}{30}=\frac{5.2}{13}$

$13x=156$

$x=12$

12

### Exercise #10

The triangle ABC is similar to the triangle DEF.

The ratio between the lengths of their sides is 9:8.

What is the ratio between the areas of the triangles?

### Step-by-Step Solution

We multiply the ratio by 2

$9:8=18:16$

Raised to the power of 2:

$9^2:8^2=81:64$

81:64

### Exercise #11

If the ratio of the areas of similar triangles is 1:16, and the length of the side of the larger triangle is 42 cm, what is the length of the corresponding side in the smaller triangle?

### Step-by-Step Solution

The ratio of similarity is 1:4

The length of the corresponding side in the small triangle is:

$\frac{42}{4}=6$

10.5

### Exercise #12

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?

### Step-by-Step Solution

Let's call the small triangle A and the large triangle B, let's write the ratio:

$\frac{A}{B}=\frac{3}{4}$

Square it:

$\frac{S_A}{S_B}=(\frac{3}{4})^2$

$\frac{S_A}{S_B}=\frac{9}{16}$

Therefore, the ratio is 9:16

9:16

### Exercise #13

In similar triangles, the area of the triangles is 361 cm² and 81 cm². If it is known that the perimeter of the first triangle is 38, what is the perimeter of the second triangle?

### Step-by-Step Solution

To begin with we can determine the perimeter of the second triangle by using the equation below.

$\frac{P_2}{P_1}=\sqrt{\frac{S_2}{S_1}}$

We insert the existing data

$\frac{P_2}{38}=\sqrt{\frac{81}{361}}$

$\frac{P_2}{38}=\frac{\sqrt{81}}{\sqrt{361}}=\frac{9}{19}$

Lastly we multiply by 38 to obtain the following answer:

$P_2=\frac{9}{19}\times38=18$

18

### Exercise #14

ABCD is a parallelogram
BFCE is a deltoid

What is the area of the parallelogram ABCD?

### Step-by-Step Solution

First, we must remember the formula for the area of a parallelogram:$\text{Lado }x\text{ Altura}$.

In this case, we will try to find the height CH and the side BC.

We start from the side

First, let's observe the small triangle EBG,

As it is a right triangle, we can use the Pythagorean theorem (

$A^2+B^2=C^2$)

$BG^2+4^2=5^2$

$BG^2+16=25$

$BG^2=9$

$BG=3$

Now, let's start looking for GC.

First, remember that the deltoid has two pairs of equal adjacent sides, therefore:$FC=EC=9$

Now we can also do Pythagoras in the triangle GCE.

$GC^2+4^2=9^2$

$GC^2+16=81$

$GC^2=65$

$GC=\sqrt{65}$

Now we can calculate the side BC:

$BC=BG+GT=3+\sqrt{65}\approx11$

Now, let's observe the triangle BGE and DHC

Angle BGE = 90°
Angle CHD = 90°
Angle CDH=EBG because these are opposite parallel angles.

Therefore, there is a ratio of similarity between the two triangles, so:

$\frac{HD}{BG}=\frac{HC}{GE}$

$\frac{HD}{BG}=\frac{7.5}{3}=2.5$

$\frac{HC}{EG}=\frac{HC}{4}=2.5$

$HC=10$

Now that there is a height and a side, all that remains is to calculate.

$10\times11\approx110$

### Answer

$\approx110$

### Exercise #15

Given that triangles ABC and DEF are similar, what is their ratio of similarity?

5:4

### Topics learned in later sections

1. Similarity of Geometric Figures