A A A B B B C C C D D D E E E 60° 30° 30° 60° $$ ΔACB∼ΔBED $$ Choose the correct answer.

$$ \frac{CB}{BE}=\frac{AC}{ED} $$

$$ \frac{AB}{BD}=\frac{ED}{AC} $$

What is the scale factor between the two triangles below? 15 15 15 30 30 30 10 10 10 5 5 5 A A A B B B C C C D D D E E E F F F

$$ \frac{FD}{CA}=\frac{EF}{CB}=\frac{ED}{AB} $$

$$ \frac{AB}{BC}=\frac{ED}{DF}=\frac{AC}{EF} $$

What is the ratio of similarity between the two triangles in the diagram below? 3m 3m 3m 2.5m 2.5m 2.5m 2m 2m 2m A A A B B B C C C D D D

There are no similar triangles in the figure.

The Ratio of Similarity: Identifying and defining elements

Question 1

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.

Question 2

According to which theorem are the triangles similar?

What is their ratio of similarity?

Question 3

$$ ΔACB\simΔBED $$

Choose the correct answer.

Question 4

The triangles above are similar.

Calculate the perimeter of the larger triangle.

Question 5

The triangle above are similar.

What is the perimeter of the blue triangle?

Examples with solutions for The Ratio of Similarity: Identifying and defining elements

Exercise #1

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.

Video Solution

Step-by-Step Solution

Let's look at the order of letters of the triangles that match each other and see the ratio of the sides.

We will write accordingly:

Triangle ABC is similar to triangle DFE

The order of similarity ratio will be:

$\frac{AB}{DF}=\frac{BC}{FE}=\frac{AC}{DE}$

Now let's insert the existing data we have in the diagram:

$\frac{8y}{9y}=\frac{7m}{FE}$

Let's reduce y and we get:

$\frac{8}{9}FE=7m$

$FE=\frac{9}{8}\times7m$

$FE=7\frac{7}{8}m$

Answer

$7\frac{7}{8}m$

Exercise #2

According to which theorem are the triangles similar?

What is their ratio of similarity?

Video Solution

Step-by-Step Solution

Using the given data, the side ratios can be written as follows:

$\frac{FD}{AB}=\frac{X}{2X}=\frac{1}{2}$

$\frac{FE}{AC}=\frac{\frac{y}{2}}{y}=\frac{y}{2y}=\frac{1}{2}$

$\frac{DE}{BC}=\frac{2Z}{4Z}=\frac{2}{4}=\frac{1}{2}$

We can therefore deduce that the ratio is compatible with the S.S.S theorem (Side-Side-Side):

$\frac{FD}{AB}=\frac{FE}{AC}=\frac{DE}{BC}=\frac{1}{2}$

Answer

S.S.S., $\frac{1}{2}$

Exercise #3

$ΔACB∼ΔBED$

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.

$\frac{AB}{DB}$

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.

$\frac{AC}{ED}$

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

$\frac{CB}{EB}$

Therefore, from this it can be deduced that:

$\frac{AB}{BD}=\frac{AC}{ED}$

And also:

$\frac{CB}{ED}=\frac{AB}{BD}$

Answer

Answers a + b are correct.

Exercise #4

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):

$3.5+1.5+4=9$

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 $x$ :

$\frac{x}{9}=\frac{14}{3.5}$

$3.5x=14\times9$

$3.5x=126$

$x=36$

Answer

Exercise #5

The triangle above are similar.

What is the perimeter of the blue triangle?

Video Solution

Step-by-Step Solution

The perimeter of the left triangle: 13+12+5=25+5=30

Therefore, the perimeter of the right triangle divided by 30 is equal to 5.2 divided by 13:

$\frac{x}{30}=\frac{5.2}{13}$

$13x=156$

$x=12$

Answer

Question 1

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?

Question 2

If the ratio of the areas of similar triangles is 1:16, and the length of the side of the larger triangle is 42 cm, what is the length of the corresponding side in the smaller triangle?

Question 3

BC is parallel to DE.

Fill in the gap:

$\frac{AD}{}=\frac{AE}{AC}$

Question 4

What is the ratio of similarity between the triangles shown in the diagram below?

Question 5

According to which theorem are the triangles congruent in the diagram?

Complete the similarity ratio:

$\frac{AB}{DF}=\frac{BC}{}=\frac{}{EF}$

Exercise #6

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?

Video Solution

Step-by-Step Solution

Let's call the small triangle A and the large triangle B, let's write the ratio:

$\frac{A}{B}=\frac{3}{4}$

Square it:

$\frac{S_A}{S_B}=(\frac{3}{4})^2$

$\frac{S_A}{S_B}=\frac{9}{16}$

Therefore, the ratio is 9:16

Answer

9:16

Exercise #7

If the ratio of the areas of similar triangles is 1:16, and the length of the side of the larger triangle is 42 cm, what is the length of the corresponding side in the smaller triangle?

Video Solution

Step-by-Step Solution

The ratio of similarity is 1:4

The length of the corresponding side in the small triangle is:

$\frac{42}{4}=6$

Answer

10.5

Exercise #8

BC is parallel to DE.

Fill in the gap:

$\frac{AD}{}=\frac{AE}{AC}$

Video Solution

Answer

Exercise #9

What is the ratio of similarity between the triangles shown in the diagram below?

Video Solution

Answer

$\frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD}$

Exercise #10

According to which theorem are the triangles congruent in the diagram?

Complete the similarity ratio:

$\frac{AB}{DF}=\frac{BC}{}=\frac{}{EF}$

Video Solution

Answer

S.A.S.
$AC=2,DE=1$

Question 1

Complete the similarity ratio given that the triangles below are similar:

$\frac{AB}{}=\frac{}{EF}=\frac{AC}{}$

Question 2

What is the scale factor between the two triangles below?

Question 3

Given that triangles ABC and DEF are similar, what is their ratio of similarity?

Question 4

According to which theorem are the triangles below similar?

What is their ratio of similarity?

Answer

$\frac{2}{3}$

Question 1

$\frac{AB}{ED}=\frac{BC}{FD}$

What is the scale factor?

Question 2

What is the ratio of similarity between the triangles below?

Question 3

According to which theorem are the triangles similar? What is the ratio of similarity?

Question 4

What is the scale factor between the triangles below?

Answer

5:12