Similar Triangles Analysis: Comparing Triangles with Sides 4 and 5 Units

SSS Similarity with Congruent Triangles

Are the triangles below similar?

555444444555444444AAABBBCCCDDDEEEFFF

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Are the triangles similar?
00:03 We want to find the similarity ratio
00:07 This is the ratio of sides
00:10 If all side ratios are equal, then the triangles are similar
00:14 This pair of sides ratio is equal
00:19 This ratio is also equal
00:24 The similarity ratio is equal, therefore the triangles are similar
00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Are the triangles below similar?

555444444555444444AAABBBCCCDDDEEEFFF

2

Step-by-step solution

To solve this problem, we'll determine if the triangles ABC\triangle ABC and DEF\triangle DEF are similar using the Side-Side-Side (SSS) similarity criterion.

Step 1: Identify the sides of both triangles:
For ABC\triangle ABC, the side lengths are AB=5AB = 5, BC=4BC = 4, and CA=4CA = 4.
For DEF\triangle DEF, the side lengths are DE=5DE = 5, EF=4EF = 4, and FD=4FD = 4.

Step 2: Calculate the ratios of the corresponding sides:
ABDE=55=1\frac{AB}{DE} = \frac{5}{5} = 1
BCEF=44=1\frac{BC}{EF} = \frac{4}{4} = 1
CAFD=44=1\frac{CA}{FD} = \frac{4}{4} = 1

Step 3: Verify similarity:
All three ratios are equal, so by the SSS criterion, the triangles are similar.

Therefore, the triangles ABC\triangle ABC and DEF\triangle DEF are similar.

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic
  • SSS Criterion: All corresponding side ratios must be equal
  • Technique: Calculate 55=1 \frac{5}{5} = 1 and 44=1 \frac{4}{4} = 1 for each pair
  • Check: When all ratios equal 1, triangles are congruent (special similarity) ✓

Common Mistakes

Avoid these frequent errors
  • Assuming triangles are similar without calculating ratios
    Don't just look at side lengths and guess = wrong conclusions! Even if triangles look similar, you must calculate the actual ratios of corresponding sides. Always compute each ratio: side₁/side₂ to verify they're all equal.

Practice Quiz

Test your knowledge with interactive questions

1027.51.5The two parallelograms above are similar. The ratio between their sides is 3:4.

What is the ratio between the the areas of the parallelograms?

FAQ

Everything you need to know about this question

What does it mean when all the ratios equal 1?

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When all ratios equal 1, it means the triangles are congruent! Congruent triangles are a special case of similar triangles where the scale factor is 1 (same size and shape).

How do I know which sides correspond to each other?

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Match sides by position and length. In this problem, both triangles have sides of 5, 4, and 4, so the 5's correspond, and the 4's correspond to each other.

Do I need to check angles too?

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No! The SSS criterion says if all three side ratios are equal, the triangles are automatically similar. You don't need to measure or calculate angles.

What if the ratios were different numbers but all equal?

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The triangles would still be similar! For example, if all ratios equaled 23 \frac{2}{3} , the triangles would be similar with a scale factor of 23 \frac{2}{3} .

Can triangles be similar if only two ratios are equal?

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No! For SSS similarity, you need all three corresponding side ratios to be equal. If even one ratio is different, the triangles are not similar.

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