Similarity ratio

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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

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Test yourself on similarity ratio!

einstein

AAABBBCCCDDDEEE60°30°30°60°\( ΔACB∼ΔBED \)

Choose the correct answer.

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We will exemplify this through an exercise.

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

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 A is equal and equal to the angle K K and the angle B B is equal to the angle L L .

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

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

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

We obtain:

KLAB=63=2{KL \over AB} = {6 \over 3} = 2

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


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Similarity ratio exercises

Exercise 1

Task

Given:

ΔACBΔBED ΔACB∼ΔBED

Choose the correct answer

Exercise 1 Given ΔACB∼ΔBED.

Solution

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

Answer

Answers a+b a + b


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Exercise 2

Task

Similar triangles:

Exercise 2 Similar triangles

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

Solution

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

ABDE=ACDF=BCEF \frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}

105=2 \frac{10}{5}=2

Answer

2 2


Exercise 3

Question

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

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

Solution

From image comes out:

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

Answer

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


Exercise 4:

Task

Choose the correct answer

Similarity ratio

Solution

Ratio of similarity

ADAF=14 \frac{AD}{AF}=\frac{1}{4}

AFAB=14 \frac{AF}{AB}=\frac{1}{4}

Answer

ADAF=AFAB \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 ΔDEF and ΔABC ΔABC .

Exercise 5 Find the similarity ratio corresponding to the triangles ΔDEF and ΔABC.

Solution

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

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

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

It follows that

2575=13 \frac{25}{75}=\frac{1}{3}

BCFE=ACFD=ABDE=13 \frac{BC}{FE}=\frac{AC}{FD}=\frac{AB}{DE}=\frac{1}{3}

Answer

13 \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. ABCDEF \triangle ABC\sim\triangle DEF

Calculate the similarity ratio

The ratio of similarity is obtained by dividing the corresponding sides of two similar figures

Given that ABCDEF \triangle ABC\sim\triangle DEF

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

A=D \sphericalangle A=\sphericalangle D

B=E \sphericalangle B=\sphericalangle E

Then the corresponding sides are AB AB ,DE DE

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

ABDE=1210=65=1.2 \frac{AB}{DE}=\frac{12}{10}=\frac{6}{5}=1.2

Therefore the similarity ratio is 1.2 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

ABCDEFGH ABCD\sim EFGH

Calculate the similarity ratio

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 AD ,EH EH , then we can calculate the similarity ratio.

EHAD=104=52=2.5 \frac{EH}{AD}=\frac{10}{4}=\frac{5}{2}=2.5

Therefore the similarity ratio is 2.5 2.5


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