Analyzing Similarity Ratios in Nested Triangles: Ratio = 1 Investigation

Triangle Similarity with Nested Configurations

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

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Step-by-step written solution

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1

Understand the problem

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

2

Step-by-step solution

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.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Similarity Ratio: Ratio of 1 means triangles are congruent (identical)
  • Visual Analysis: Compare corresponding side lengths in nested triangles
  • Verification: Check if all corresponding sides have equal ratios ✓

Common Mistakes

Avoid these frequent errors
  • Assuming nested triangles have ratio 1
    Don't assume triangles inside each other have similarity ratio 1 = congruent triangles! This ignores their clearly different sizes. Always compare the actual corresponding side lengths to determine the true ratio.

Practice Quiz

Test your knowledge with interactive questions

If it is known that both triangles are equilateral, are they therefore similar?

FAQ

Everything you need to know about this question

What does similarity ratio of 1 actually mean?

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A similarity ratio of 1 means the triangles are congruent - they have exactly the same size and shape! This is much stronger than just being similar.

How can I tell if triangles are similar without measurements?

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Visual inspection can help! If triangles have the same angles but clearly different sizes (like nested triangles), they're similar but not congruent.

Can similar triangles ever have different similarity ratios?

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Each pair of similar triangles has one specific similarity ratio. If you're comparing three triangles, you get three different ratios: A:B, B:C, and A:C.

Why can't nested triangles have ratio 1?

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Look at the diagram! The outer triangle is clearly larger than the inner ones. If the ratio were 1, all triangles would be exactly the same size - which they're not!

What's the difference between similar and congruent?

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Similar: same shape, possibly different size. Congruent: same shape and same size. Congruent triangles are similar with ratio = 1.

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