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 3 , we know that each side of the large triangle is 3 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.

A1 - How do we calculate the similarity ratio

Suggested Topics to Practice in Advance

  1. Similar Triangles
  2. Triangle similarity criteria
  3. Similarity of Triangles and Polygons

Practice The Ratio of Similarity

Examples with solutions for The Ratio of Similarity

Exercise #1

AAABBBCCCDDDEEE60°30°30°60°ΔACBΔBED ΔACB∼ΔBED

Choose the correct answer.

Video Solution

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.

ABDB \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.

ACED \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

CBEB \frac{CB}{EB}

Therefore, from this it can be deduced that:

ABBD=ACED \frac{AB}{BD}=\frac{AC}{ED}

And also:

CBED=ABBD \frac{CB}{ED}=\frac{AB}{BD}

Answer

Answers a + b are correct.

Exercise #2

AAABBBCCCMMMNNN36 What is the ratio between the sides of the triangles ΔABC and ΔMNA?

Video Solution

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:

ABAM=BCMN=ACAN \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 MN=3,BC=6 63=2 \frac{6}{3}=2

Answer

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

Exercise #3

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.

Answer

No

Exercise #4

BC is parallel to DE.

Fill in the gap:

AD=AEAC \frac{AD}{}=\frac{AE}{AC}

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Since we are given that line BC is parallel to DE

Angle E equals angle C and angle D equals angle B - corresponding angles between parallel lines are equal.

Now let's observe that angle D is opposite to side AE and angle B is opposite to side AC, meaning:

AEAC \frac{AE}{AC}

Now let's observe that angle E is opposite to side AD and angle C is opposite to side AB, meaning:

ADAB \frac{AD}{AB}

Answer

AB

Exercise #5

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.8y8y8y7m7m7m9y9y9yAAABBBCCCDDDEEEFFF

Video Solution

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:

ABDF=BCFE=ACDE \frac{AB}{DF}=\frac{BC}{FE}=\frac{AC}{DE}

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

8y9y=7mFE \frac{8y}{9y}=\frac{7m}{FE}

Let's reduce y and we get:

89FE=7m \frac{8}{9}FE=7m

FE=98×7m FE=\frac{9}{8}\times7m

FE=778m FE=7\frac{7}{8}m

Answer

778m 7\frac{7}{8}m

Exercise #6

What is the ratio of similarity between the triangles shown in the diagram below?

AAABBBDDDCCCEEE

Video Solution

Step-by-Step Solution

From the drawing it appears that angle E equals angle A

Since angle D equals 90 degrees, its adjacent angle also equals 90 degrees.

In other words, angle D1 equals angle D2 and both equal 90 degrees.

Since we have two pairs of equal angles, the triangles are similar.

Also angle B equals angle C

Now let's write the similar triangles according to their corresponding angle letters:

ABC=ECD ABC=ECD

Let's write the ratio of sides according to the corresponding letters of the similar triangles:

ABEC=ADED=BDCD \frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD}

Answer

ABEC=ADED=BDCD \frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD}

Exercise #7

According to which theorem are the triangles similar?

What is their ratio of similarity?

2x2x2x4z4z4zyyy2z2z2zxxxAAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Using the given data, the side ratios can be written as follows:

FDAB=X2X=12 \frac{FD}{AB}=\frac{X}{2X}=\frac{1}{2}

FEAC=y2y=y2y=12 \frac{FE}{AC}=\frac{\frac{y}{2}}{y}=\frac{y}{2y}=\frac{1}{2}

DEBC=2Z4Z=24=12 \frac{DE}{BC}=\frac{2Z}{4Z}=\frac{2}{4}=\frac{1}{2}

We can therefore deduce that the ratio is compatible with the S.S.S theorem (Side-Side-Side):

FDAB=FEAC=DEBC=12 \frac{FD}{AB}=\frac{FE}{AC}=\frac{DE}{BC}=\frac{1}{2}

Answer

S.S.S., 12 \frac{1}{2}

Exercise #8

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.

888444666999333666333111222AAABBBCCCGGGHHHIIIDDDEEEFFFABC

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:

GHDE=HIEF=GIDF \frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}

96=31=62=3 \frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3

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

Answer

a a and b b , similarity ratio of 3 3

Exercise #9

3.51.54146

The triangles above are similar.

Calculate the perimeter of the larger triangle.

Video Solution

Step-by-Step Solution

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

3.5+1.5+4=9 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 x :

x9=143.5 \frac{x}{9}=\frac{14}{3.5}

3.5x=14×9 3.5x=14\times9

3.5x=126 3.5x=126

x=36 x=36

Answer

36

Exercise #10

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

Video Solution

Step-by-Step Solution

We square it. 7 squared is equal to 49.

Answer

49

Exercise #11

BC is parallel to DE.

Calculate AE.

151515101010222444666AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Let's prove that triangles ADE and ABC are similar using:

Since DE is parallel to BC, angles ADE and ABC are equal (according to the law - between parallel lines, corresponding angles are equal)

Angle DAE and angle BAC are equal since it's the same angle

After we proved that the triangles are similar, let's write the given data from the drawing according to the following similarity ratio:

ADAB=DEBC=AEAC \frac{AD}{AB}=\frac{DE}{BC}=\frac{AE}{AC}

We know that - AB=AD+DB=6+4=10 AB=AD+DB=6+4=10

610=10154=AEAC \frac{6}{10}=\frac{10}{154}=\frac{AE}{AC}

Let's reduce the fractions:

35=23 \frac{3}{5}=\frac{2}{3}

This statement is incorrect, meaning the data in the drawing contradicts the fact that the triangles are similar. Therefore, the drawing is impossible.

Answer

Impossible as the shape in the figure cannot exist.

Exercise #12

5.213125 The triangle above are similar.

What is the perimeter of the blue triangle?

Video Solution

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:

x30=5.213 \frac{x}{30}=\frac{5.2}{13}

13x=156 13x=156

x=12 x=12

Answer

12

Exercise #13

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?

1021.57.5

Video Solution

Step-by-Step Solution

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

AB=34 \frac{A}{B}=\frac{3}{4}

Square it:

SASB=(34)2 \frac{S_A}{S_B}=(\frac{3}{4})^2

SASB=916 \frac{S_A}{S_B}=\frac{9}{16}

Therefore, the ratio is 9:16

Answer

9:16

Exercise #14

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?

Video Solution

Step-by-Step Solution

The ratio of similarity is 1:4

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

424=6 \frac{42}{4}=6

Answer

10.5

Exercise #15

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?

Video Solution

Step-by-Step Solution

We multiply the ratio by 2

9:8=18:16 9:8=18:16

Raised to the power of 2:

92:82=81:64 9^2:8^2=81:64

Answer

81:64

Topics learned in later sections

  1. Similarity of Geometric Figures