Triangle Similarity Theorem: Finding the Ratio between ABC and DEF

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

Look at the vertex labels and side positions. In triangles ABC and DEF, side AB corresponds to DE, side AC corresponds to DF, and side BC corresponds to EF. Match vertices in the same order.

Q: What if only two ratios are equal?

Then the triangles are not similar! For SSS similarity, all three corresponding side ratios must be exactly equal. Even if two match, you need all three.

Q: Can I use different similarity theorems for the same triangles?

Yes, but you only need one theorem to prove similarity! If SSS works (all side ratios equal), that's sufficient proof. You don't need to check AA or SAS unless SSS fails.

Q: What does the 1/2 ratio tell me about the triangles?

It means triangle DEF is half the size of triangle ABC in all linear dimensions. The areas would have a ratio of $ (\frac{1}{2})^2 = \frac{1}{4} $.

Triangle Similarity with Side Ratios

According to which theorem are the triangles similar?

What is their ratio of similarity?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 According to which theorem are the triangles similar? And what is the similarity ratio?

00:03 Let's find the similarity ratio between the sides

00:09 Let's substitute appropriate values according to the given data and find the ratio

00:12 This is the ratio, if it's equal for all sides then the triangles are similar

00:16 Let's check another pair of sides

00:25 The ratio matches

00:30 And let's check the last pair of sides

00:38 Here too the ratio matches

00:45 The triangles are similar according to SSS

00:49 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

According to which theorem are the triangles similar?

What is their ratio of similarity?

Step-by-step solution

Using the given data, the side ratios can be written as follows:

$\frac{FD}{AB}=\frac{X}{2X}=\frac{1}{2}$

$\frac{FE}{AC}=\frac{\frac{y}{2}}{y}=\frac{y}{2y}=\frac{1}{2}$

$\frac{DE}{BC}=\frac{2Z}{4Z}=\frac{2}{4}=\frac{1}{2}$

We can therefore deduce that the ratio is compatible with the S.S.S theorem (Side-Side-Side):

$\frac{FD}{AB}=\frac{FE}{AC}=\frac{DE}{BC}=\frac{1}{2}$

Final Answer

S.S.S., $\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

SSS Theorem: All three corresponding side ratios must be equal
Ratio Calculation: $\frac{x}{2x} = \frac{2z}{4z} = \frac{y/2}{y} = \frac{1}{2}$
Verification: Check all three ratios simplify to same value ✓

Common Mistakes

Avoid these frequent errors

Confusing which triangle is larger
Don't assume triangle ABC is larger because it's named first = wrong ratio direction! This gives you 2:1 instead of 1:2. Always compare corresponding sides systematically: smaller triangle to larger triangle.

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

How do I know which sides correspond to each other?

Look at the vertex labels and side positions. In triangles ABC and DEF, side AB corresponds to DE, side AC corresponds to DF, and side BC corresponds to EF. Match vertices in the same order.

Why is the ratio 1/2 instead of 2:1?

The ratio depends on which triangle you put first! Since triangle DEF has smaller sides (x, 2z, y/2) compared to triangle ABC (2x, 4z, y), the ratio is $\frac{\text{smaller}}{\text{larger}} = \frac{1}{2}$ .

What if only two ratios are equal?

Then the triangles are not similar! For SSS similarity, all three corresponding side ratios must be exactly equal. Even if two match, you need all three.

Can I use different similarity theorems for the same triangles?

Yes, but you only need one theorem to prove similarity! If SSS works (all side ratios equal), that's sufficient proof. You don't need to check AA or SAS unless SSS fails.

What does the 1/2 ratio tell me about the triangles?

It means triangle DEF is half the size of triangle ABC in all linear dimensions. The areas would have a ratio of $(\frac{1}{2})^2 = \frac{1}{4}$ .

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