The Ratio of Similarity: Applying the formula

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$

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.

The key property of similar triangles is that the ratios of the lengths of their corresponding sides are equal. For triangles EDB and ABC:

• Triangle EDB is similar to triangle ABC, meaning:
• Corresponding sides are proportional: $\frac{AB}{BE} = \frac{BC}{DB} = \frac{AC}{ED}$

Therefore, the correct relationship among the sides, using the concept of similar triangles, is:

$\frac{BC}{DB} = \frac{AB}{BE} = \frac{AC}{ED}$

This matches choice 3.

Thus, the correct answer to the problem is $\frac{BC}{DB}=\frac{AB}{BE}=\frac{AC}{ED}$.

To determine if the triangles $\triangle ABC$ and $\triangle KLT$ are similar, we apply the Side-Side-Side (SSS) similarity criterion. This requires that the ratios of corresponding sides are equal.

We are given the side lengths: $BC = 12$, $AB = 9$, $CA = 6$ for $\triangle ABC$, and $LK = 2$, $KT = 3$, $LT = 4$ for $\triangle KLT$.

First, find the ratio for each pair of corresponding sides:

• Compare $BC$ and $LT$: $\frac{BC}{LT} = \frac{12}{4} = 3$
• Compare $CA$ and $LK$: $\frac{CA}{LK} = \frac{6}{2} = 3$
• Compare $AB$ and $KT$: $\frac{AB}{KT} = \frac{9}{3} = 3$

Since $\frac{BC}{LT} = \frac{CA}{LK} = \frac{AB}{KT} = 3$, all sides maintain a constant ratio. Hence, the triangles are similar.

The similarity ratio is $3$, indicating $\triangle ABC \sim \triangle KLT$ with a ratio of 3:1.

The correct choice, as given in the options, is:

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

To find the correct proportionality relationship between the triangles ADE and ABC, we need to identify the corresponding sides:

• In triangle ADE, the sides $AD$, $AE$, and $DE$ correspond to triangle ABC’s sides $AB$, $AC$, and $BC$, respectively.

Because these triangles are congruent, the ratios of their corresponding sides are equal:

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

This relationship matches with choice 1:

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

Therefore, the correct answer is choice 1.

To solve the problem, we need to determine the ratio of lengths between sides $AB$ and $DE$ in triangles $\triangle ABC$ and $\triangle CDE$, using their similarity.

Given:

• Triangle $\triangle ABC$ and triangle $\triangle CDE$ are similar by AA criterion, having common angle $C$ and both right triangles.
• The vertical height from $B$ to $A$ is $3$, while the vertical height from $D$ to $C$ is $12$.
• The horizontal length from $A$ to $C$ is $2$, and from $C$ to $E$ is $8$.

To find the similarity ratio, we can compare corresponding segments:

• The vertical height ratio is $\frac{BD}{DE} = \frac{3}{12} = \frac{1}{4}$.
• The horizontal base ratio is $\frac{AC}{CE} = \frac{2}{8} = \frac{1}{4}$.

Thus, the triangles are similar with a ratio of $\frac{1}{4}$.

Since all corresponding dimensions of similar triangles are proportional by this ratio, it follows:

• $\frac{AB}{DE} = \frac{1}{4}$.

Therefore, the solution to the problem is: $\frac{AB}{DE} = \frac{1}{4}$.

The correct answer choice is: $\frac{1}{4}$.

To solve the problem, we proceed with the following steps:

• Identify the given side lengths for each triangle.
• Compare the side ratios of each triangle pair to check for similarity.
• Verify the side ratios to affirm the similarity ratio.
• Select the correct multiple-choice answer based on the analysis.

Given side lengths:
Triangle C: $6$, $3$, $3$ (perpendicular and base, as seen in figure).
Triangle B: $4.5$, $3$, $2$ (perpendicular and base, as seen in figure).
Triangle A: $6$, $4$, $3.5$ (perpendicular and base, as seen in figure).

Calculating the ratios:

• For triangles C and B:
  $\frac{6}{4.5} = \frac{3}{2}$ which simplifies to $\frac{3}{2} = 1.5$, indicating that triangles C and B are similar.
• Comparison for other pairs: Triangle A with Triangle B or C reveals no common proportionality.

Therefore, the only pair of similar triangles is C and B with a similarity ratio of $\frac{3}{2}$ or 1.5.

The correct choice is, therefore, C + B are similar with a ratio of 1.5.

To solve this problem, we'll compare the side ratios of the given triangles to determine which pair are similar and find the similarity ratio.

• Step 1: Identify the given triangle side lengths:
  Triangle A: Sides of length $8$, $4$, and $6$.
  Triangle B: Sides of length $10$, $6$, and $8$.
  Triangle C: Sides of length $4$, $2$, and $3$.
• Step 2: Compare the ratios of corresponding sides between pairs of triangles.

Comparing Triangle A and Triangle B:

• Ratio $\frac{8}{10} = 0.8$; $\frac{4}{6} \approx 0.67$; $\frac{6}{8} = 0.75$

Here, the ratios are not equal; hence, triangles A and B are not similar.

Comparing Triangle A and Triangle C:

• Ratio $\frac{8}{4} = 2$; $\frac{4}{2} = 2$; $\frac{6}{3} = 2$

All ratios are equal, so triangles A and C are similar, with a similarity ratio of 2.

Comparing Triangle B and Triangle C:

• Ratio $\frac{10}{4} = 2.5$; $\frac{6}{2} = 3$; $\frac{8}{3} \approx 2.67$

The ratios are not equal, so triangles B and C are not similar.

Therefore, the similar triangles are Triangle A and Triangle C, with a similarity ratio of 2.

The correct answer is A + C are similar with a ratio of 2.

To solve this problem, we need to calculate the similarity ratio between $\Delta GH F$ and $\Delta ABC$. Since both triangles are equilateral:

• The side length of $\Delta ABC$ is $18$, and the side length of $\Delta GH F$ is $2$.
• The similarity ratio $\frac{\text{side of } \Delta ABC}{\text{side of } \Delta GHF}$ is:
\item $\frac{18}{2} = 9.$

Therefore, the similarity ratio between triangles $\Delta GHF$ and $\Delta ABC$ is $9$. The correct choice is:

$\frac{AB}{GF}=\frac{AC}{FH}=9$.

To solve this problem, we need to use the properties of similar triangles. It is given that triangles DBC and ABC are similar. When triangles are similar, it means that their corresponding sides are proportional.

For triangles DBC and ABC, considering the similarity, the corresponding sides should satisfy:

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

The ratio $\frac{BC}{BC}$ simplifies to 1, which is a characteristic of the proportional relationship between the sides of similar triangles.

To solve for the correct choice, let's compare this with the options provided:

• Option 1: $\frac{AD}{AB} = \frac{AE}{AD} = \frac{BC}{BC}$ - This does not align because segments AD and AE aren't involved in the main similarity relation.
• Option 2: $\frac{AB}{BD} = \frac{AC}{CD} = \frac{BC}{BC}$ - This perfectly aligns with our derived relationship.
• Option 3: $\frac{AD}{DB} = \frac{AE}{EC} = \frac{BC}{BC}$ - This option has segments that are not directly part of the triangles ABC and DBC as described.
• Option 4: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{BC}{BC}$ - Again, this involves segments not described in the initial triangle similarity.

Therefore, the correct correspondence that mathematically represents the similarity of the given triangles is found in Option 2.

Hence, the correct relation of similarity is: $\frac{AB}{BD} = \frac{AC}{CD} = \frac{BC}{BC}$.

To solve this problem, we will use the properties of similar triangles. In similar triangles, the corresponding sides are proportional. This means if triangles ADE and ABC are similar, the ratios of these corresponding sides must be equal. We can set up the proportion by considering:

• Side $AD$ in triangle ADE corresponds to side $AB$ in triangle ABC.
• Side $AE$ in triangle ADE corresponds to side $AC$ in triangle ABC.
• Side $DE$ in triangle ADE corresponds to side $BC$ in triangle ABC.

Therefore, based on this correspondence, the ratio of the sides should be:

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

To confirm, let's ensure we are interpreting the similarity correctly:

• Triangle ADE is similar to triangle ABC, meaning their corresponding angles are equal, and sides track on a consistent ratio.

Thus, based on the properties of similar triangles, the correct answer corresponding to these proportion relationships is:

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

This corresponds to choice 4.

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.

Triangles ADE and ABC are given as congruent. When two triangles are congruent, their corresponding sides are equal in proportion. Therefore, we need to find the correct proportional relationship between the sides of these triangles.

Let's recall that for two congruent triangles, their corresponding sides are in equal ratios. Specifically, for triangles ADE and ABC, the sides AD, AE, and DE correspond to the sides AB, AC, and BC, respectively.

Thus, the correct proportional relationship between the corresponding sides of triangles ADE and ABC is:

• $\frac{AD}{AB}$ corresponding to the first set of sides.
• $\frac{AE}{AC}$ corresponding to the second set of sides.
• $\frac{DE}{BC}$ corresponding to the third set of sides.

Therefore, the choice that correctly represents the proportional relationship of the sides of triangles ADE and ABC is:

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

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.

This problem involves identifying the correct similarity ratio of triangles based on given line segments. We have triangle $\triangle ACF$ with line segments $AC$ and $CF$, and triangle $\triangle BDV$ with line segments $BD$ and $DV$, where corresponding side lengths of similar triangles should satisfy proportional relationships.

First, recognize that for similar triangles, the ratio of corresponding sides should be equal. If triangles $\triangle ACF$ and $\triangle BDV$ are similar, the segments would meet certain proportional criteria. The possible ratios could be formed by recognizing:

• $\frac{FV}{AC}$ should correspond to the smaller segment extending from a vertex to base or equivalent in the other triangle setup.
• $\frac{DV}{AB}$ matches up the vertex downward extension similar to $\frac{FV}{AC}$ with a base side.
• $\frac{FD}{BC}$ must be the ratio of a cross-cutting or diagonal side ratio to maintain similarity from base to opposite corner.

Thus, the correct setup for these segments should reflect that:
$\frac{FV}{AC} = \frac{DV}{AB} = \frac{FD}{BC}$

By evaluating each choice given, the correct answer would align with this reasoning. Therefore, the correct choice is: $\frac{FV}{AC} = \frac{DV}{AB} = \frac{FD}{BC}$.

To find the ratio of similarity between triangles $\triangle BCD$ and $\triangle ABC$, we start by noting the configuration of rectangle $ABCD$.

Since $ABCD$ is a rectangle, $\angle ABC$ and $\angle BCD$ are right angles. Thus, triangles $\triangle BCD$ and $\triangle ABC$ are right triangles.

Both triangles share the same height (side length $BC = 5$) and base (in triangle $\triangle BCD, \ DC = 9$ and in triangle $\triangle ABC, \ AB = 9$).

The important observation is that despite differing configurations, these triangles maintain a proportionate structure, both sharing the same dimensions in the rectangle. This can make both triangles similar.

Thus, the ratio of similarity between the sides of $\triangle BCD$ and $\triangle ABC$ is 1.

Therefore, the solution to the problem is 1.

To solve this problem, let's follow these steps:

• Step 1: Identify similar triangles and related segments.
• Step 2: Apply proportions based on geometric similarity principles.
• Step 3: Compare with the given options.

Let's work through each step:

Step 1: From the geometry, consider triangles and line segments such as $AD$, $AF$, and $AB$. These segments often form part of similar triangles with known properties.

Step 2: Based on triangle similarity properties, the ratio for segments in similar figures should follow a coherent pattern like that of proportions described for viable geometric figures, namely: $\frac{AD}{AF} = \frac{AF}{AB}$ This indicates the relationship of subsegments generated by these points. Use triangle proportionality theorem or similar property for derivation.

Step 3: Match this with the answer selections provided. The choice: $\frac{AD}{AF} = \frac{AF}{AB}$ corresponds directly with option 3 among the provided choices.

Therefore, the correct answer is: $\frac{AD}{AF}=\frac{AF}{AB}$.

To solve this problem, let's go through the steps clearly:

• Step 1: Since triangles $\triangle ABC$ and $\triangle DEF$ are equilateral, each side of these triangles is equal to the other sides of the same triangle.
• Step 2: Moreover, given that $\triangle ABC \cong \triangle DEF$, these triangles are congruent, meaning that the corresponding sides and angles are equal. Therefore, side $AB = DE$, $BC = EF$, and $CA = FD$.

Since every side of $\triangle ABC$ is equal to every corresponding side of $\triangle DEF$, the ratio between any corresponding sides of these triangles is:

Therefore, all sides have a ratio of $1$, which means that the correct statement is that "The ratio between the sides is equal to 1."

Therefore, the correct choice is choice 4.

Hence, the solution to this problem is: The ratio between the sides is equal to 1.

We square it. 7 squared is equal to 49.

We multiply the ratio by 2

$9:8=18:16$

Raised to the power of 2:

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