The Ratio of Similarity - Examples, Exercises and Solutions

Understanding The Ratio of Similarity

Complete explanation with examples

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

Detailed explanation

Practice The Ratio of Similarity

Test your knowledge with 19 quizzes

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

Examples with solutions for The Ratio of Similarity

Step-by-step solutions included
Exercise #1

AAABBBCCCDDDEEE60°30°30°60°ΔACBΔBED 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.

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.

Video Solution
Exercise #2

AAABBBCCCMMMNNN36 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:

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

Video Solution
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

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

Video Solution
Exercise #5

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.8y8y8y7m7m7m9y9y9yAAABBBCCCDDDEEEFFF

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

Video Solution

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