Similar triangles are triangles whose three angles are equal respectively and also the ratio between each pair of corresponding sides is equal. Two similar triangles are actually larger or smaller versions each other.
The ratio of similarity is the ratio between two corresponding sides in two similar triangles.
To prove similarities betweentriangles, we will use the following theorems:
Angle-Angle (A.A): If two angles are equal respectively between two triangles, then the triangles are similar.
Side-Angle-Side (S.A.S): If the ratio of two pairs of sides is equal, and also the angles between them are equal to each other, then the triangles are similar.
Side-Side-Side (S.S.S.): If for two triangles, the ratio of the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio), then the triangles are similar.
For similarity of polygons we will define it this way: if for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar.
Intuitively, just like similar triangles, also two similar polygons are actually an enlargement or reduction of each other.
The two parallelograms above are similar. The ratio between their sides is 3:4.
What is the ratio between the the areas of the parallelograms?
Incorrect
Correct Answer:
9:16
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Similar triangles
Definition: Similar triangles are triangles whose three angles are equal respectively and also the ratio of each pair of corresponding sides is equal. Two similar triangles are actually an enlargement or reduction of each other.
To understand this, let's look at the following example:
Example 1
Given the two triangles in the drawing
ΔABC ΔDEF
Given that the triangle ΔABC and the triangle ΔDEF are similar triangles. We will mark this with the sign
It looks like this: ΔABC ~ It is important to write the correct order of the vertices, similar to the superposition of triangles. ΔDEF
From here we can conclude that the three angles are equal respectively, ie:
∢A=∢D ∢B=∢E ∢C=∢F
And we can conclude that the ratio between each pair of corresponding sides is equal. That is:
DEAB=EFBC=FDCA
This ratio of sides is called the ratio of similarities. It is important to note that two overlapping triangles are also similar triangles when the ratio of similarity is 1.
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Test your knowledge
Question 1
The ratio of the areas of similar triangles is \( \frac{9}{100} \)Given that the perimeter of the large triangle is 129 cm, what is the perimeter of the small triangle?
Incorrect
Correct Answer:
38.7
Question 2
What is the ratio between the sides of the triangles ΔABC and ΔMNA?
Incorrect
Correct Answer:
\( \frac{BC}{MN}=2 \)
Question 3
Is the similarity ratio between the three triangles equal to one?
Incorrect
Correct Answer:
No
What is the similarity ratio?
The similarity ratio is the ratio between two corresponding sides in two similar triangles.
Example 2
Since the two triangles ΔABC and ΔDEF are similar triangles, that is:
ΔABC ~ ΔDEF
It is also given AB=8 BC=12 CA=6 In addition: DE=4 EF=6 FD=3 All the data are marked in the drawing.
Calculate the similarity ratio between the two triangles.
Pay attention that we do not know the size of the angles, but we do not need it to calculate the similarity ratio, since it is stated that they are similar triangles, so their respective angles measure the same. We can calculate the similarity ratio by the ratio between each pair of corresponding sides:
DEAB=48=2 EFBC=612=2 FDCA=36=2
That is, we have seen that the similarity ratio between the triangle ΔABC For the triangle ΔDEF Is 1:2. QED
Pay attention to the similarity ratio between the lengths of the sides of the triangle ΔDEF For the triangle ΔABC It is 2:1 Intuitively, the length of each side in a triangle ΔABC is 2 times longer than each side in a triangle ΔDEF Respectively.
To prove the similarity between triangles we will use one of the following three theorems: - Angle-Angle (A.A): If two angles are equal in correspondence between two triangles, then the triangles are similar. - Side-Angle-Side (S.A.S): If the ratio between two pairs of sides is equal, and also the angles included between them are equal to each other, then the triangles are similar. - Side-side-side (S.S.S.): If for two triangles, the ratio between the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio) then the triangles are similar.
Exercise with example - (How to calculate the length of the side)
Given two triangles in the drawing below ΔABC ΔDEF are similar triangles, i.e. ΔABC ~ ΔDEF In addition given: AB=5 DE=2.5 FD=1 ∢A=∢D ∢B=∢E ∢C=∢F
All data are labeled in the drawing.
Question: what is the length of the side AC?
Solution: The two triangles are similar, so we will calculate the similarity ratio and use it to solve the assignment. Remember that the ratio between two sides in similar triangles is equal and therefore:
DEAB=2.55=12
That is, the similarity ratio is 2:1, and each side of the triangle △ABC is twice as large as any corresponding side in the triangle △DEF. Now we can calculate the length of side AC. According to the similarity ratio:
DFAC=2
We replace and obtain:
1AC=2
That is, we obtained:
AC=2
QED
Do you know what the answer is?
Question 1
BC is parallel to DE.
Fill in the gap:
\( \frac{AD}{}=\frac{AE}{AC} \)
Incorrect
Correct Answer:
AB
Question 2
Triangle DFE is similar to triangle ABC.
Calculate the length of FE.
Incorrect
Correct Answer:
\( 7\frac{7}{8}m \)
Question 3
What is the ratio of similarity between the triangles shown in the diagram below?
Incorrect
Correct Answer:
\( \frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD} \)
Similar polygons
Definition: If for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar. Intuitively, as in similar triangles, two similar polygons are actually larger and smaller versions of each other.
Example 3 - Similar polygons
These two squares are similar squares:
Any two corresponding angles are equal since all angles are equal. The ratio of the two corresponding sides, i.e., the similarity ratio, is 2/1 or, in other words, each of the sides is twice as large for the large square as for the small square.
Check your understanding
Question 1
According to which theorem are the triangles congruent in the diagram?
Complete the similarity ratio:
\( \frac{AB}{DF}=\frac{BC}{}=\frac{}{EF} \)
Incorrect
Correct Answer:
S.A.S. \( AC=2,DE=1 \)
Question 2
According to which theorem are the triangles similar?
What is their ratio of similarity?
Incorrect
Correct Answer:
S.S.S., \( \frac{1}{2} \)
Question 3
Complete the similarity ratio given that the triangles below are similar:
\( \frac{AB}{}=\frac{}{EF}=\frac{AC}{} \)
Incorrect
Correct Answer:
\( DE=1,BC=2,DF=3 \)
Example 4 - Similar polygons
Two pentagons in the drawing are similar, which means that any pair of corresponding angles are equal. When the similarity ratio is
ABFG=23=11.5
That is, for each pair of corresponding sides, the length of the pentagon FGHIJ is 1.5 times greater than that of the pentagon ABCDE.
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Yes, according to A.A the two triangles are similar, because since they are equilateral triangles, then their three angles measure the same.
Answer
Yes
Do you know what the answer is?
Question 1
The ratio of the areas of similar triangles is \( \frac{9}{100} \)Given that the perimeter of the large triangle is 129 cm, what is the perimeter of the small triangle?
Incorrect
Correct Answer:
38.7
Question 2
What is the ratio between the sides of the triangles ΔABC and ΔMNA?
Incorrect
Correct Answer:
\( \frac{BC}{MN}=2 \)
Question 3
Is the similarity ratio between the three triangles equal to one?
Incorrect
Correct Answer:
No
Review questions
What is similarity of triangles?
Two triangles are similar if their three respective angles have the same measure and the ratio between the pairs of their respective sides is the same.
What are the three criteria for similarity of triangles?
There are three criteria to determine if two triangles are similar or not, which are the following:
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.
How to find the similarity ratio?
In order to get the ratio of similarity we must calculate the ratio or division of each of their respective pairs of sides, and this relationship is the same for each of the pairs of sides when two triangles are similar.
When are two triangles similar by the SSS criterion?
Two triangles are similar when they have the same shape but their sides do not necessarily have to measure the same, they are similar by the SSS criterion as long as their corresponding sides are proportional, i.e. the ratio between their sides is the same.
What is the similarity of polygons?
In the same way that the similarity of triangles, two polygons are similar when they have the same shape, they do not necessarily have to have the same measure in their sides, that is, their corresponding angles are equal and the ratio of their corresponding sides is the same for all of them.
Check your understanding
Question 1
BC is parallel to DE.
Fill in the gap:
\( \frac{AD}{}=\frac{AE}{AC} \)
Incorrect
Correct Answer:
AB
Question 2
Triangle DFE is similar to triangle ABC.
Calculate the length of FE.
Incorrect
Correct Answer:
\( 7\frac{7}{8}m \)
Question 3
What is the ratio of similarity between the triangles shown in the diagram below?
Incorrect
Correct Answer:
\( \frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD} \)
Examples with solutions for Similar Triangles and Polygons
Exercise #1
Are the two triangles similar?
Video Solution
Step-by-Step Solution
To find out if the triangles are similar, we can check if there is an appropriate similarity ratio between their sides.
The similarity ratio is the constant difference between the corresponding sides.
In this case, we can check if:
5062.5=80100=80100
5062.5=100125=110025=141
80100=810=142=141
Therefore:141=141=141
Therefore, we can say that there is a constant ratio of141 between the sides of the triangles and therefore the triangles are similar.
Answer
Yes
Exercise #2
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.
DBAB
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.
EDAC
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
EBCB
Therefore, from this it can be deduced that:
BDAB=EDAC
And also:
EDCB=BDAB
Answer
Answers a + b are correct.
Exercise #3
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:
AMAB=MNBC=ANAC
Now we can calculate the ratio between the sides of the given triangles:
MN=3,BC=636=2
Answer
MNBC=2
Exercise #4
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 #5
The two parallelograms above are similar. The ratio between their sides is 3:4.
What is the ratio between the the areas of the parallelograms?
Video Solution
Step-by-Step Solution
The square of the ratio between the sides is equal to the ratio between the areas of the parallelograms: