Similar Triangles Analysis: Finding the Similarity Ratio Among Three Given Triangles

Similar Triangle Identification with Ratio Calculations

In the image there are a pair of similar triangles and a triangle that is not similar to the others.

Determine which are similar and calculate their similarity ratio.

888444666999333666333111222AAABBBCCCGGGHHHIIIDDDEEEFFFABC

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

In the image there are a pair of similar triangles and a triangle that is not similar to the others.

Determine which are similar and calculate their similarity ratio.

888444666999333666333111222AAABBBCCCGGGHHHIIIDDDEEEFFFABC

2

Step-by-step solution

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:

GHDE=HIEF=GIDF \frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}

96=31=62=3 \frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3

That is, the ratio between them is 1:3.

3

Final Answer

a a and b b , similarity ratio of 3 3

Key Points to Remember

Essential concepts to master this topic
  • SSS Similarity: Three pairs of corresponding sides must be proportional
  • Ratio Check: 96=31=62=3 \frac{9}{6} = \frac{3}{1} = \frac{6}{2} = 3 confirms similarity
  • Verification: All three ratios equal the same value confirms similar triangles ✓

Common Mistakes

Avoid these frequent errors
  • Comparing sides in wrong order
    Don't match sides randomly like comparing triangle A's longest side to triangle B's shortest side = wrong ratios! This gives inconsistent proportions and wrong conclusions. Always identify corresponding sides by position and angle relationships first.

Practice Quiz

Test your knowledge with interactive questions

Angle B is equal to 60°

Angle C is equal to 55°

Angle E is equal to 60°

Angle F is equal to 50°

Are these triangles similar?

AAABBBCCCDDDEEEFFF

FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

+

Look at the position of each side in relation to the vertices. The sides opposite similar angles correspond, and sides between similar angle pairs also correspond.

What if only two ratios are equal?

+

Then the triangles are not similar! For SSS similarity, all three ratios must be exactly equal. Even if two match perfectly, one different ratio means no similarity.

Can the similarity ratio be a decimal?

+

Absolutely! Ratios like 1.5, 2.5, or 0.75 are perfectly valid. Just make sure all three ratios equal the same decimal value for the triangles to be similar.

Why isn't triangle C similar to the others?

+

Let's check: Triangle A to C gives ratios 83 \frac{8}{3} , 41 \frac{4}{1} , and 62 \frac{6}{2} . Since these don't equal each other, triangle C is not similar to triangles A or B.

Does the order of triangles matter when finding ratios?

+

Yes! Always be consistent - if you write Triangle A over Triangle B for the first ratio, use the same order for all three ratios. Mixing orders will give you wrong results.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Similar Triangles and Polygons questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations