Are the triangles below similar? If so, what is their ratio?AAABBBCCCKKKLLLTTT6912342

Yes, similarity ratio:$ \frac{AC}{KT}=\frac{BC}{LT}=\frac{AB}{LK} $

Yes, similarity ratio:$ \frac{BC}{LK}=\frac{AB}{LT}=\frac{AC}{KT} $

Similar Triangles: Calculate the Ratio Between ABC (12,9,6) and KLT (4,3,2)

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

Look at the side lengths and match them by size! The longest side of one triangle corresponds to the longest side of the other. Here: BC=12 (longest) matches LT=4 (longest), CA=6 (shortest) matches LK=2 (shortest).

Q: What if I get different ratios for each pair of sides?

If the ratios are different, then the triangles are not similar! For similarity, all three ratios must be exactly equal. Even small differences mean the triangles don't have the same shape.

Q: Does the order of the ratio matter?

Yes! Always write the ratio as larger triangle : smaller triangle consistently. Here we use $ \frac{ABC}{KLT} $ giving ratios of 3, not $ \frac{KLT}{ABC} $ which would give ratios of 1/3.

Q: Why is the similarity ratio 3:1 and not just 3?

The ratio 3:1 shows the relationship clearly - triangle ABC is 3 times larger than triangle KLT. We can also write it as just "3" or as the fraction $ \frac{3}{1} $.

Q: Can I use this method for any triangle similarity problem?

SSS similarity works when you know all three side lengths of both triangles. If you only know some sides or angles, you'll need different methods like SAS or AA similarity.

Q: What happens if triangles are similar but oriented differently?

Orientation doesn't matter! Similar triangles can be rotated, flipped, or positioned differently and still be similar. Focus on matching corresponding sides by length, not by position in the diagram.

SSS Similarity Criterion with Ratio Calculations

Are the triangles below similar? If so, what is their ratio?

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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 Let's find the ratio of corresponding sides

00:09 If the ratio of all sides is equal, then the triangles are similar

00:13 Let's substitute appropriate values according to the given data and find the similarity ratio

00:29 The ratio of all sides is equal, therefore the triangles are similar

00:34 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

Are the triangles below similar? If so, what is their ratio?

Step-by-step solution

To determine if the triangles $\triangle ABC$ and $\triangle KLT$ are similar, we apply the Side-Side-Side (SSS) similarity criterion. This requires that the ratios of corresponding sides are equal.

We are given the side lengths: $BC = 12$ , $AB = 9$ , $CA = 6$ for $\triangle ABC$ , and $LK = 2$ , $KT = 3$ , $LT = 4$ for $\triangle KLT$ .

First, find the ratio for each pair of corresponding sides:

Compare $BC$ and $LT$ : $\frac{BC}{LT} = \frac{12}{4} = 3$
Compare $CA$ and $LK$ : $\frac{CA}{LK} = \frac{6}{2} = 3$
Compare $AB$ and $KT$ : $\frac{AB}{KT} = \frac{9}{3} = 3$

Since $\frac{BC}{LT} = \frac{CA}{LK} = \frac{AB}{KT} = 3$ , all sides maintain a constant ratio. Hence, the triangles are similar.

The similarity ratio is $3$ , indicating $\triangle ABC \sim \triangle KLT$ with a ratio of 3:1.

The correct choice, as given in the options, is:

^{Yes, similarity ratio:}
$\frac{BC}{LT}=\frac{CA}{LK}=\frac{AB}{KT}$

Final Answer

^{Yes, similarity ratio:}
$\frac{BC}{LT}=\frac{CA}{LK}=\frac{AB}{KT}$

Key Points to Remember

Essential concepts to master this topic

SSS Similarity Rule: All three pairs of corresponding sides must have equal ratios
Technique: Calculate each ratio: $\frac{12}{4} = \frac{6}{2} = \frac{9}{3} = 3$
Check: All ratios equal 3, so triangles are similar with scale factor 3:1 ✓

Common Mistakes

Avoid these frequent errors

Comparing sides in wrong order
Don't match sides randomly like BC with LK = 12/2 = 6 and AB with LT = 9/4 = 2.25! These unequal ratios suggest no similarity when triangles actually are similar. Always identify corresponding sides carefully by comparing side lengths in order.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

Look at the side lengths and match them by size! The longest side of one triangle corresponds to the longest side of the other. Here: BC=12 (longest) matches LT=4 (longest), CA=6 (shortest) matches LK=2 (shortest).

What if I get different ratios for each pair of sides?

If the ratios are different, then the triangles are not similar! For similarity, all three ratios must be exactly equal. Even small differences mean the triangles don't have the same shape.

Does the order of the ratio matter?

Yes! Always write the ratio as larger triangle : smaller triangle consistently. Here we use $\frac{ABC}{KLT}$ giving ratios of 3, not $\frac{KLT}{ABC}$ which would give ratios of 1/3.

Why is the similarity ratio 3:1 and not just 3?

The ratio 3:1 shows the relationship clearly - triangle ABC is 3 times larger than triangle KLT. We can also write it as just "3" or as the fraction $\frac{3}{1}$ .

Can I use this method for any triangle similarity problem?

SSS similarity works when you know all three side lengths of both triangles. If you only know some sides or angles, you'll need different methods like SAS or AA similarity.

What happens if triangles are similar but oriented differently?

Orientation doesn't matter! Similar triangles can be rotated, flipped, or positioned differently and still be similar. Focus on matching corresponding sides by length, not by position in the diagram.

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Similar Triangles: Calculate the Ratio Between ABC (12,9,6) and KLT (4,3,2)

SSS Similarity Criterion with Ratio Calculations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which sides correspond to each other?

What if I get different ratios for each pair of sides?

Does the order of the ratio matter?

Why is the similarity ratio 3:1 and not just 3?

Can I use this method for any triangle similarity problem?

What happens if triangles are similar but oriented differently?

🌟 Unlock Your Math Potential

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