Similar Triangles: Comparing ABC (7,5,4) and DEF (7,5,3) Triangles

Q: Why aren't these triangles similar when two sides are the same?

For triangles to be similar, all corresponding side ratios must be equal. Even though sides 7 and 5 match perfectly, the third sides (4 vs 3) create different ratios: $ \frac{4}{3} \neq 1 $.

Q: Do I always need to check all three side ratios?

Yes! The SSS similarity criterion requires all three ratios to be equal. If even one ratio is different, the triangles are not similar.

Q: What if I get ratios like 2:2:2 versus 1:1:1?

Those triangles would be similar! The actual values don't matter - what matters is that all ratios are equal. $ \frac{2}{1} = \frac{2}{1} = \frac{2}{1} = 2 $ means similar triangles.

Q: What does it mean when ratios are not equal?

Unequal ratios mean the triangles have different shapes. They might share some side lengths, but they're not scaled versions of each other - so they're not similar.

Triangle Similarity with Unequal Side Ratios

Are triangles below similar?

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Step-by-step video solution

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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

00:15 This pair of sides ratio is equal

00:21 This pair of sides ratio is not equal

00:26 The similarity ratio is not equal, therefore the triangles are not similar

00:36 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Are triangles below similar?

Step-by-step solution

To determine whether the triangles are similar, we will use the Side-Side-Side (SSS) criterion for similarity. According to this criterion, triangles are similar if the ratios of their corresponding sides are equal.

We have two triangles: $\triangle ABC$ with sides 7, 5, and 4, and $\triangle DEF$ with sides 7, 5, and 3.

We will calculate the ratios of the corresponding sides:

For sides $AB$ and $DE$ : $\frac{AB}{DE} = \frac{7}{7} = 1$
For sides $BC$ and $EF$ : $\frac{BC}{EF} = \frac{5}{5} = 1$
For sides $AC$ and $DF$ : $\frac{AC}{DF} = \frac{4}{3}$

From the calculations, we observe that two of the side ratios are equal to 1, but the third ratio $\frac{4}{3}$ does not match the others. Thus, the side ratios are not all identical, meaning the triangles are not similar according to the SSS criterion.

Therefore, the triangles $\triangle ABC$ and $\triangle DEF$ are not similar.

Final Answer

Key Points to Remember

Essential concepts to master this topic

SSS Similarity Rule: All three side ratios must be equal for similar triangles
Ratio Method: Calculate $\frac{7}{7} = 1$ , $\frac{5}{5} = 1$ , $\frac{4}{3}$ and compare
Verification: Check that 1 = 1 ≠ $\frac{4}{3}$ proves triangles not similar ✓

Common Mistakes

Avoid these frequent errors

Assuming triangles are similar because some sides match
Don't think triangles with sides 7,5,4 and 7,5,3 are similar because two sides match = wrong conclusion! Partial matches mean nothing in similarity. Always check that ALL three ratios are identical: $\frac{7}{7} = \frac{5}{5} = \frac{4}{3}$ must be true.

Practice Quiz

Test your knowledge with interactive questions

The 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

Why aren't these triangles similar when two sides are the same?

For triangles to be similar, all corresponding side ratios must be equal. Even though sides 7 and 5 match perfectly, the third sides (4 vs 3) create different ratios: $\frac{4}{3} \neq 1$ .

Do I always need to check all three side ratios?

Yes! The SSS similarity criterion requires all three ratios to be equal. If even one ratio is different, the triangles are not similar.

What if I get ratios like 2:2:2 versus 1:1:1?

Those triangles would be similar! The actual values don't matter - what matters is that all ratios are equal. $\frac{2}{1} = \frac{2}{1} = \frac{2}{1} = 2$ means similar triangles.

How do I organize the ratios correctly?

Match corresponding sides carefully! List sides in order (like shortest to longest) for both triangles, then create ratios:

Triangle 1 sides: 4, 5, 7
Triangle 2 sides: 3, 5, 7
Ratios: $\frac{4}{3}, \frac{5}{5}, \frac{7}{7}$

What does it mean when ratios are not equal?

Unequal ratios mean the triangles have different shapes. They might share some side lengths, but they're not scaled versions of each other - so they're not similar.

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