The Ratio of Similarity: Identifying and defining elements

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:

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

From the drawing it appears that angle E equals angle A

Since angle D equals 90 degrees, its adjacent angle also equals 90 degrees.

In other words, angle D1 equals angle D2 and both equal 90 degrees.

Since we have two pairs of equal angles, the triangles are similar.

Also angle B equals angle C

Now let's write the similar triangles according to their corresponding angle letters:

$ABC=ECD$

Let's write the ratio of sides according to the corresponding letters of the similar triangles:

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

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:

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

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

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

Since we are given that line BC is parallel to DE

Angle E equals angle C and angle D equals angle B - corresponding angles between parallel lines are equal.

Now let's observe that angle D is opposite to side AE and angle B is opposite to side AC, meaning:

$\frac{AE}{AC}$

Now let's observe that angle E is opposite to side AD and angle C is opposite to side AB, meaning:

$\frac{AD}{AB}$

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):

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

To determine which theorem proves the triangles are similar, we'll use the Angle-Angle (AA) Similarity Theorem:

• Step 1: Check the angles $\angle ABC$ and $\angle DEF$, and $\angle ACB$ and $\angle DFE$.
• Step 2: Since the problem implies these angles are equal, the AA criterion confirms the triangles are similar.

Next, we calculate the ratio of similarity:

• Step 3: Identify the corresponding sides, such as $AB$ and $ED$, $BC$ and $DF$, and $AC$ and $EF$.
• Step 4: Establish the correct ratio:
  $\frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF}$

Therefore, according to the AA similarity theorem, the triangles are similar with the ratio of similarity $\frac{AB}{ED}=\frac{BC}{DF}=\frac{AC}{EF}$.

The correct choice is:

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

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 $x$:

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

$3.5x=14\times9$

$3.5x=126$

$x=36$

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

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:

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

$13x=156$

$x=12$

The ratio of similarity is 1:4

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

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

To solve this problem, follow these steps:

• Step 1: Find the ratio of similarity between the corresponding sides of the triangles $\triangle DEF$ and $\triangle ABC$.
• Step 2: Simplify the ratio of the sides.
• Step 3: Apply the ratio to find the perimeter of $\triangle DEF$.

Let's follow these steps:

Step 1: Given $EF = 18$ and $BC = 24$ are corresponding sides of similar triangles $\triangle DEF \sim \triangle ABC$, we find the ratio:

$\frac{EF}{BC} = \frac{18}{24}$

Step 2: Simplify this ratio:

$\frac{18}{24} = \frac{3}{4}$

Step 3: The ratio of similarity $\frac{DE}{AB} = \frac{EF}{BC} = \frac{3}{4}$ means the perimeter of $\triangle DEF$ is $\frac{3}{4}$ of the perimeter of $\triangle ABC$.

Given the perimeter of $\triangle ABC$ is 64, compute the perimeter of $\triangle DEF$:

$\frac{3}{4} \times 64 = 48$

Therefore, the perimeter of $\triangle DEF$ is 48.