AAABBBCCCDDD15157 Are triangles ΔABD and ΔADC similar?

No, we do not know the length of the third side.

They are not necessarily similar.

Triangle Similarity Proof: Comparing Triangles ABD and ADC with Heights of 7 and 15 Units

Q: Why does the height of 7 matter if we're using SSS?

The height of 7 units shows that AD is perpendicular to BC, which is a key property of isosceles triangles. This confirms that AD bisects the base BC into equal segments.

Q: How do we know BD = DC without measurements?

In an isosceles triangle, when you draw a height from the apex (vertex A) to the base BC, it always bisects the base into two equal segments. This is a fundamental property!

Q: What's the difference between congruent and similar triangles?

Congruent triangles are identical in size and shape (SSS proves congruence). Similar triangles have the same shape but may differ in size. Since these triangles are congruent, they're automatically similar too!

Q: Why don't the side lengths of 15 units affect the similarity?

The 15 units represent the equal sides AB and AC of the isosceles triangle. Since both triangles share these same sides, it actually supports the SSS similarity criterion.

Q: Can I use a different similarity test instead of SSS?

You could also use SAS (Side-Angle-Side) since both triangles share the right angle at D, and have AD common plus either AB = AC. However, SSS is most direct here.

Isosceles Triangle Similarity with Height Properties

Are triangles ΔABD and ΔADC similar?

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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 The triangle is isosceles according to the given data

00:08 In an isosceles triangle, the height is also a median (T)

00:15 Legs are equal in an isosceles triangle (T)

00:23 Common side between the triangles (T)

00:29 The triangles are congruent by SAS

00:35 Congruent triangles are necessarily similar

00:38 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 triangles ΔABD and ΔADC similar?

Step-by-step solution

Since side AB is equal to side AC, the triangle is isosceles.

AD divides BC into two equal parts, therefore BD = DC.

Also, AD is a common side.

From this, it follows that the triangles are similar according to the S.S.S (side side side) criterion.

Final Answer

Yes, according to S.S.S.

Key Points to Remember

Essential concepts to master this topic

Isosceles Rule: Equal sides create equal base segments when height drawn
SSS Criterion: AB = AC, BD = DC, AD = AD (common side)
Verification: Three corresponding sides equal proves triangles congruent and similar ✓

Common Mistakes

Avoid these frequent errors

Assuming triangles aren't similar without checking all sides
Don't conclude triangles aren't similar just because heights are different = missing the isosceles property! Different heights don't affect similarity when sides match. Always identify if AB = AC first, then check if this creates equal base segments BD = DC.

Practice Quiz

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FAQ

Everything you need to know about this question

Why does the height of 7 matter if we're using SSS?

The height of 7 units shows that AD is perpendicular to BC, which is a key property of isosceles triangles. This confirms that AD bisects the base BC into equal segments.

How do we know BD = DC without measurements?

In an isosceles triangle, when you draw a height from the apex (vertex A) to the base BC, it always bisects the base into two equal segments. This is a fundamental property!

What's the difference between congruent and similar triangles?

Congruent triangles are identical in size and shape (SSS proves congruence). Similar triangles have the same shape but may differ in size. Since these triangles are congruent, they're automatically similar too!

Why don't the side lengths of 15 units affect the similarity?

The 15 units represent the equal sides AB and AC of the isosceles triangle. Since both triangles share these same sides, it actually supports the SSS similarity criterion.

Can I use a different similarity test instead of SSS?

You could also use SAS (Side-Angle-Side) since both triangles share the right angle at D, and have AD common plus either AB = AC. However, SSS is most direct here.

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