Similarity of triangles and polygons

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 between triangles, 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.

Image 1 similar triangles

Practice Similar Triangles and Polygons

Exercise #1

10062.5508080100 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:

62.550=10080=10080 \frac{62.5}{50}=\frac{100}{80}=\frac{100}{80}

62.550=125100=125100=114 \frac{62.5}{50}=\frac{125}{100}=1\frac{25}{100}=1\frac{1}{4}

10080=108=124=114 \frac{100}{80}=\frac{10}{8}=1\frac{2}{4}=1\frac{1}{4}

 Therefore:114=114=114 1\frac{1}{4}=1\frac{1}{4}=1\frac{1}{4}

Therefore, we can say that there is a constant ratio of114 1\frac{1}{4} between the sides of the triangles and therefore the triangles are similar.

Answer

Yes

Exercise #2

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 #3

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 #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

1027.51.5The 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:

32:42=9:16 3^2:4^2=9:16

Answer

9:16

Exercise #1

Angle B is equal to 40°

Angle C is equal to 60°

Angle E is equal to 40°

Angle F is equal to 60°

Are the triangles similar?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Given that the data shows that there are two pairs with equal angles:

B=E=40 B=E=40

C=F=60 C=F=60

The triangles are similar according to the angle-angle theorem, therefore triangle ABC is similar to triangle DEF.

Answer

Yes

Exercise #2

Angle B is equal to 70 degrees

Angle C is equal to 35 degrees

Angle E is equal to 70 degrees

Angle F is equal to 35 degrees

Are the triangles similar?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

The triangles are similar according to the angle-angle theorem.

Having two pairs of equal angles is sufficient to conclude that the triangles are similar.

Answer

Yes

Exercise #3

AAADDDFFFBBBCCC65°40°40°Which of the following are true?

Video Solution

Step-by-Step Solution

If the sum of the angles in a triangle equals 180, then angle F equals 75 and therefore angle C also equals 75.

The triangles are similar according to the angle-angle theorem

Answer D is correct

Answer

Answers (a) and (b) are correct.

Exercise #4

AAABBBCCCDDDFFFVVV694.580°60°80°40°436 Are the triangles above similar?

Video Solution

Step-by-Step Solution

Answer

Answers b and c are correct.

Exercise #5

Square A is greater than square B by a ratio of 23 \frac{2}{3} .

If the perimeter of square A is known to be 56, what is the area of square B?

Video Solution

Step-by-Step Solution

We will mark the side in square A as X

Therefore the perimeter will be:
4x=56 4x=56

x=14 x=14

Now we can calculate the area of square A:

14×14=196 14\times14=196

As we are given the ratio between the areas:

196S2=(23)2=49 \frac{196}{S_2}=(\frac{2}{3})^2=\frac{4}{9}

That is, the ratio will be:

196X=49 \frac{196}{X}=\frac{4}{9}

The area of the square will be equal to:

9×1964=17644=441 \frac{9\times196}{4}=\frac{1764}{4}=441

Answer

441

Exercise #1

Are the triangles below similar?

666999888555999888AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

The sides of the triangles are not equal and, therefore, the triangles are not similar.

Answer

No

Exercise #2

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 #3

Are the two triangles similar?

If so, according to which theorem?

121212999444333AAABBBCCCDDDEEEFFF25°25°

Video Solution

Step-by-Step Solution

Since we neither have data on sides AB and DE nor the other angles, it is impossible to know.

Answer

It is impossible to know.

Exercise #4

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 #5

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