Triangle Similarity Investigation: Comparing Two Triangles with Sides 100, 80, and 62.5

Q: Why do I need to check all three side ratios?

For SSS (Side-Side-Side) similarity, all three corresponding sides must have the exact same ratio. If even one ratio is different, the triangles are not similar!

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

Match sides by relative size: shortest with shortest, middle with middle, longest with longest. In this problem: 50↔62.5, 80↔100, 80↔100.

Q: Is it easier to use decimals or fractions for ratios?

Both work! $ \frac{62.5}{50} = 1.25 $ or $ 1\frac{1}{4} $ are equivalent. Use whichever form makes the comparison clearer for you.

Q: What if the triangles have the same angles but different side ratios?

That's impossible! If two triangles have the same angles (AAA), their corresponding sides must have the same ratio. This is a fundamental property of similar triangles.

Q: Can triangles be similar if only two ratios match?

No! For similarity, you need either: all three side ratios equal (SSS), two angles equal (AA), or two sides proportional with included angle equal (SAS).

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 ratios between the sides, if the ratio is equal then they are similar

00:07 Let's divide each side by its corresponding side

00:14 This is the similarity ratio that should also result from another pair of sides

00:18 Let's find the ratio of sides in the other pairs

00:21 In this pair, the ratio of sides is equal

00:27 And in this pair too, the ratio of sides is equal

00:31 The similarity ratio is equal, therefore the triangles are similar

00:35 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 two triangles similar?

2

Step-by-step solution

To find out if the triangles are similar, we can check if there is an appropriate similarity ratio between their sides.

The similarity ratio is the constant difference between the corresponding sides.

In this case, we can check if:

$\frac{62.5}{50}=\frac{100}{80}=\frac{100}{80}$

$\frac{62.5}{50}=\frac{125}{100}=1\frac{25}{100}=1\frac{1}{4}$

$\frac{100}{80}=\frac{10}{8}=1\frac{2}{4}=1\frac{1}{4}$

Therefore: $1\frac{1}{4}=1\frac{1}{4}=1\frac{1}{4}$

Therefore, we can say that there is a constant ratio of $1\frac{1}{4}$ between the sides of the triangles and therefore the triangles are similar.

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

SSS Similarity: All three corresponding sides must have same ratio
Technique: Calculate ratios $\frac{62.5}{50} = \frac{100}{80} = 1.25$
Check: Convert to mixed numbers: $1\frac{1}{4} = 1\frac{1}{4} = 1\frac{1}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Checking only two side ratios instead of all three
Don't compare just two ratios like 62.5/50 = 100/80 and assume similarity! This misses the third side comparison and can give false conclusions. Always verify all three ratios: first/first = second/second = third/third.

Practice Quiz

Test your knowledge with interactive questions

FAQ

Everything you need to know about this question

Why do I need to check all three side ratios?

+

For SSS (Side-Side-Side) similarity, all three corresponding sides must have the exact same ratio. If even one ratio is different, the triangles are not similar!

How do I know which sides correspond to each other?

+

Match sides by relative size: shortest with shortest, middle with middle, longest with longest. In this problem: 50↔62.5, 80↔100, 80↔100.

Is it easier to use decimals or fractions for ratios?

+

Both work! $\frac{62.5}{50} = 1.25$ or $1\frac{1}{4}$ are equivalent. Use whichever form makes the comparison clearer for you.

What if the triangles have the same angles but different side ratios?

+

That's impossible! If two triangles have the same angles (AAA), their corresponding sides must have the same ratio. This is a fundamental property of similar triangles.

Can triangles be similar if only two ratios match?

+

No! For similarity, you need either: all three side ratios equal (SSS), two angles equal (AA), or two sides proportional with included angle equal (SAS).

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Triangle Similarity Investigation: Comparing Two Triangles with Sides 100, 80, and 62.5

Triangle Similarity with Side Ratio Verification

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