A A A B B B C C C M M M N N N 3 6 What is the ratio between the sides of the triangles ΔABC and ΔMNA?

$$ \frac{BC}{MN}=\frac{1}{2} $$

There is no ratio between the sides.

Are the triangles below similar? If so, what is their ratio? A A A B B B C C C K K K L L L T T T 6 9 12 3 4 2

Yes, similarity ratio: $$ \frac{AC}{KT}=\frac{BC}{LT}=\frac{AB}{LK} $$

Yes, similarity ratio: $$ \frac{BC}{LK}=\frac{AB}{LT}=\frac{AC}{KT} $$

A A A B B B C C C M M M O O O N N N F F F G G G H H H 18 10 2 18 18 2 2 6 6 What is the similarity ratio between triangles ΔGHF and ΔABC?

$$ $$$$ \frac{AB}{MN}=\frac{MN}{MO}=9 $$

$$ \frac{AC}{BC}=\frac{NO}{FH}=\frac{1}{9} $$

$$ \frac{AB}{MN}=\frac{MO}{GH}=\frac{1}{9} $$

Triangles ABC and EFD are similar. Calculate the length of AB. 5 5 5 2 2 2 9 9 9 7 7 7 A A A B B B C C C D D D E E E F F F

It is not possible for the triangles to be similar as the similarity ratio is not correct.

Triangles ADE and ABC are congruent. Choose the correct answer. A A A B B B C C C D D D E E E

$$ \frac{AB}{AD}=\frac{AC}{AE}=\frac{DE}{BC} $$

$$ \frac{AD}{DB}=\frac{AE}{EC}=\frac{DE}{BC} $$

$$ \frac{AD}{AE}=\frac{AE}{DB}=\frac{DE}{BC} $$

Triangles ADE and ABC are congruent. Choose the correct answer. A A A B B B C C C D D D E E E

$$ \frac{AB}{AD}=\frac{AC}{AE}=\frac{DE}{BC} $$

$$ \frac{AD}{A}=\frac{AE}{AC}=\frac{DE}{BA} $$

The Ratio of Similarity: Applying the formula

Question 1

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

Question 2

Is the similarity ratio between the three triangles equal to one?

Question 3

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.

Question 4

BC is parallel to DE.

Calculate AE.

Question 5

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

Examples with solutions for The Ratio of Similarity: Applying the formula

Exercise #1

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:

$\frac{AB}{AM}=\frac{BC}{MN}=\frac{AC}{AN}$

Now we can calculate the ratio between the sides of the given triangles:

$MN=3,BC=6$ $\frac{6}{3}=2$

Answer

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

Exercise #2

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

Exercise #3

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.

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:

$\frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}$

$\frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3$

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

Answer

$a$ and $b$ , similarity ratio of $3$

Exercise #4

BC is parallel to DE.

Calculate AE.

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:

$\frac{AD}{AB}=\frac{DE}{BC}=\frac{AE}{AC}$

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

$\frac{6}{10}=\frac{10}{154}=\frac{AE}{AC}$

Let's reduce the fractions:

$\frac{3}{5}=\frac{2}{3}$

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 #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

Question 1

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?

Question 2

In similar triangles, the area of the triangles is 361 cm² and 81 cm². If it is known that the perimeter of the first triangle is 38, what is the perimeter of the second triangle?

Question 3

AB is parallel to CD.

Calculate the ratio between AB and DC.

Question 4

Are the triangles below similar? If so, what is their ratio?

Question 5

Given the triangle DBC similar to triangle ABC

Choose the correct answer:

Exercise #6

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

$9:8=18:16$

Raised to the power of 2:

$9^2:8^2=81:64$

Answer

81:64

Exercise #7

In similar triangles, the area of the triangles is 361 cm² and 81 cm². If it is known that the perimeter of the first triangle is 38, what is the perimeter of the second triangle?

Video Solution

Step-by-Step Solution

To begin with we can determine the perimeter of the second triangle by using the equation below.

$\frac{P_2}{P_1}=\sqrt{\frac{S_2}{S_1}}$

We insert the existing data

$\frac{P_2}{38}=\sqrt{\frac{81}{361}}$

$\frac{P_2}{38}=\frac{\sqrt{81}}{\sqrt{361}}=\frac{9}{19}$

Lastly we multiply by 38 to obtain the following answer:

$P_2=\frac{9}{19}\times38=18$

Answer

Exercise #8

AB is parallel to CD.

Calculate the ratio between AB and DC.

Video Solution

Answer

7:3

Exercise #9

Are the triangles below similar? If so, what is their ratio?

Video Solution

Answer

^{Yes, similarity ratio:}
$\frac{BC}{LT}=\frac{CA}{LK}=\frac{AB}{KT}$

Exercise #10

Given the triangle DBC similar to triangle ABC

Choose the correct answer:

Video Solution

Answer

$\frac{AB}{BD}=\frac{AC}{CD}=\frac{BC}{BC}$

Question 1

Andy: "The triangles are similar."
Gonzalo: "The scale factor is 4."
Daniel: "No, the scale factor is 2."

Who is correct?

Question 2

What is the similarity ratio between triangles ΔGHF and ΔABC?

Question 3

What is the ratio between the lengths of sides AB and DE in triangles ΔCDE and ΔABC?

Question 4

Triangle EDB is similar to triangle ABC.

Choose the correct answer.

Answer

$6\frac{3}{7}$

Question 1

Triangles ADE and ABC are congruent.

Choose the correct answer.

Question 2

Triangles ADE and ABC are congruent.

Choose the correct answer.

Question 3

Triangles ADE and ABC are similar.

Choose the appropriate answer.

Question 4

Calculate AB using the data in the diagram below.

Question 5

Answer

$13\frac{1}{3}$

Exercise #20

Calculate AC according to the data in the figure below: