The Ratio of Similarity: Identifying and defining elements

According to which theorem are the triangles similar?

What is their ratio of similarity?

According to the given data, we will write the side ratios as follows:

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

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

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

From this, it follows that the ratio is according to S.S.S (Side-Side-Side):

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

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

Triangle DFE is similar to triangle ABC.

Calculate the length of FE.

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:

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

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

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

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

$ΔACB∼ΔBED$

Choose the correct answer.

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:

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

Answers a + b are correct.

The triangles above are similar.

Calculate the perimeter of the larger triangle.

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 $x$:

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

$3.5x=14\times9$

$3.5x=126$

$x=36$

36

The triangle above are similar.

What is the perimeter of the blue triangle?

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:

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

$13x=156$

$x=12$

12

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?

The ratio of similarity is 1:4

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

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

10.5

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?

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

9:16

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

5:4

BC is parallel to DE.

Fill in the gap:

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

AB

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

Complete the similarity ratio:

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

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

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

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

$DE=1,BC=2,DF=3$

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

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

What is the scale factor between the two triangles below?

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

What is the ratio between AD and BD?

5:12

What is the ratio of similarity between the triangles below?

3:1

According to which theorem are the triangles below similar?

What is their ratio of similarity?

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

The triangles below are similar.

$\frac{BC}{EF}=\text{?}$

$2$

What is the scale factor between the triangles below?

It is not possible to determine.

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

What is the scale factor?

The triangles are not similar.

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

$\frac{2}{3}$