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.

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

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

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

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

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

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

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

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: $$ \frac{BC}{LK}=\frac{AB}{LT}=\frac{AC}{KT} $$

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.

A A A B B B C C C D D D E E E F F F J J J 24 6 1.5 Choose the correct answer.

$$ \frac{AE}{AC}=\frac{AJ}{BC} $$

$$ \frac{DE}{BC}=\frac{DE}{JF} $$

$$ \frac{JF}{BC}=\frac{DE}{BC} $$

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

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

Question 5

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?

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

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

Exercise #5

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

Question 1

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 2

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

Question 3

Triangles ADE and ABC are congruent.

Choose the correct answer.

Question 4

Triangles ADE and ABC are similar.

Choose the appropriate answer.

Question 5

Given the triangle DBC similar to triangle ABC

Choose the correct answer:

Exercise #6

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 #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

What is the similarity ratio between triangles ΔGHF and ΔABC?

Question 2

Triangle EDB is similar to triangle ABC.

Choose the correct answer.

Question 3

Triangles ADE and ABC are congruent.

Choose the correct answer.

Question 4

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

Answer

Andy and Daniel

Question 1

Triangles ABC and EFD are similar.

Calculate the length of AB.

Question 2

AB is parallel to CD.

Calculate the ratio between AB and DC.

Question 3

Choose the correct answer.

Question 4

$ΔABC≅Δ\text{DEF}$

The above triangles are equilateral.

Choose the correct answer:

Answer

The ratio between the sides is equal to 1.

Exercise #20

ABCD is a rectangle.

What is the ratio of similarity between the lengths of the sides of triangles ΔBCD and ΔABC?