AAABBBCCCMMMOOONNNFFFGGGHHH1810218182266What 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} $

Find the Similarity Ratio: Comparing Triangles ΔGHF, ΔNOM and ΔABC in a Nested Structure

Q: How do I know which sides correspond to each other?

In similar triangles, corresponding sides are in the same position. Since both triangles are equilateral, any side from ΔABC corresponds to any side from ΔGHF. Use the side lengths given in the diagram.

Q: Why does the similarity ratio matter for the order of triangles?

The ratio ΔGHF to ΔABC means ΔABC sides go in the numerator. If you flip it, you get the reciprocal! $ \frac{18}{2} = 9 $ but $ \frac{2}{18} = \frac{1}{9} $.

Q: What if the triangles weren't equilateral?

For any similar triangles, all corresponding sides must have the same ratio. You'd need to identify which vertices match up, then use any pair of corresponding sides to find the ratio.

Q: Can I use the areas instead of side lengths?

Yes, but be careful! If the side ratio is 9, then the area ratio is 9² = 81. For this problem, stick with side lengths as shown in the answer choices.

Q: How do I read nested triangles like this diagram?

Look for the labeled side lengths on each triangle. The largest triangle ΔABC has sides of 18, the middle triangle has sides of 10 and 6, and the smallest triangle ΔGHF has sides of 2.

Similarity Ratios with Nested Triangles

What is the similarity ratio between triangles ΔGHF and ΔABC?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the similarity ratio between the triangles

00:03 The equilateral triangles

00:08 In an equilateral triangle all angles are equal to 60

00:16 The triangles are similar because all angles are equal

00:33 This is the similarity ratio between the triangles

00:52 Equal sides

01:01 Let's substitute appropriate side values according to the given data

01:06 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the similarity ratio between triangles ΔGHF and ΔABC?

Step-by-step solution

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:

\frac{18}{2} = 9.

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

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Similarity ratio equals corresponding side length divided by corresponding side length
Technique: For equilateral triangles, use any side: $\frac{18}{2} = 9$
Check: Verify all corresponding sides have same ratio: AB/GF = AC/FH = BC/GH = 9 ✓

Common Mistakes

Avoid these frequent errors

Confusing which triangle is larger when writing ratios
Don't write $\frac{GF}{AB} = \frac{2}{18} = \frac{1}{9}$ when asked for ΔGHF to ΔABC ratio! This gives the reciprocal and wrong answer choices. Always put the first triangle mentioned in the numerator: $\frac{AB}{GF} = \frac{18}{2} = 9$ .

Practice Quiz

Test your knowledge with interactive questions

Is the similarity ratio between the three triangles equal to one?

FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

In similar triangles, corresponding sides are in the same position. Since both triangles are equilateral, any side from ΔABC corresponds to any side from ΔGHF. Use the side lengths given in the diagram.

Why does the similarity ratio matter for the order of triangles?

The ratio ΔGHF to ΔABC means ΔABC sides go in the numerator. If you flip it, you get the reciprocal! $\frac{18}{2} = 9$ but $\frac{2}{18} = \frac{1}{9}$ .

What if the triangles weren't equilateral?

For any similar triangles, all corresponding sides must have the same ratio. You'd need to identify which vertices match up, then use any pair of corresponding sides to find the ratio.

Can I use the areas instead of side lengths?

Yes, but be careful! If the side ratio is 9, then the area ratio is 9² = 81. For this problem, stick with side lengths as shown in the answer choices.

How do I read nested triangles like this diagram?

Look for the labeled side lengths on each triangle. The largest triangle ΔABC has sides of 18, the middle triangle has sides of 10 and 6, and the smallest triangle ΔGHF has sides of 2.

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Find the Similarity Ratio: Comparing Triangles ΔGHF, ΔNOM and ΔABC in a Nested Structure

Similarity Ratios with Nested Triangles

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which sides correspond to each other?

Why does the similarity ratio matter for the order of triangles?

What if the triangles weren't equilateral?

Can I use the areas instead of side lengths?

How do I read nested triangles like this diagram?

🌟 Unlock Your Math Potential

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