Class 10 Exam > Class 10 Questions > In an equilateral triangle ABC, if AD ⊥...
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In an equilateral triangle ABC, if AD ⊥ BC. Then
- a)3AB2 = 4AD2
- b)2AB2 = 3AD2
- c)3AB2 = 2AD2
- d)4AB2 = 3AD2
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
In an equilateral triangle ABC, if AD ⊥ BC. Thena)3AB2= 4...
∆ ABC, in which sides are AB=BC= AC= a units and AD is perpendicular to BC ,
In ∆ADB ,
AB²= AD²+ BD² (by Pythagoras theorem)
a² = AD² + (a/2)² [BD= 1/2BC, since in an equilateral triangle altitude AD is perpendicular bisector of BC ]
a²- a²/4 =AD²
⇒ ( 4a²-a²)/4 = AD²
⇒ 3a² /4 = AD²
⇒ 3AB²/4= AD² [ AB= a]
⇒ 3AB²= 4AD²
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In an equilateral triangle ABC, if AD ⊥ BC. Thena)3AB2= 4...
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In an equilateral triangle ABC, if AD ⊥ BC. Thena)3AB2= 4...
Explanation:
Given:
Equilateral triangle ABC with AD = BC
Statement to prove:
3AB^2 = 4AD^2
Proof:
1. Properties of Equilateral Triangle:
- In an equilateral triangle, all sides are equal.
- In an equilateral triangle, all angles are equal.
2. Given Information:
- AD = BC (Given)
- In an equilateral triangle, all sides are equal. Therefore, AB = BC = AC
3. Applying Pythagoras Theorem:
- In triangle ABC, applying Pythagoras Theorem, we have:
AB^2 = AD^2 + BD^2
Since AD = BC, we can substitute AD with BC:
AB^2 = BC^2 + BD^2
4. Using Properties of Equilateral Triangle:
- Since AB = BC, we can rewrite the above equation as:
AB^2 = AB^2 + BD^2
BD^2 = 0 (as AB = AB)
5. Using the Given Information:
- Since AD = BC, we have:
4AD^2 = 4BC^2
3AB^2 = 4AD^2 (substituting BC with AD)
Therefore, the statement 3AB^2 = 4AD^2 is proved. Hence, option 'A' is correct.
Given:
Equilateral triangle ABC with AD = BC
Statement to prove:
3AB^2 = 4AD^2
Proof:
1. Properties of Equilateral Triangle:
- In an equilateral triangle, all sides are equal.
- In an equilateral triangle, all angles are equal.
2. Given Information:
- AD = BC (Given)
- In an equilateral triangle, all sides are equal. Therefore, AB = BC = AC
3. Applying Pythagoras Theorem:
- In triangle ABC, applying Pythagoras Theorem, we have:
AB^2 = AD^2 + BD^2
Since AD = BC, we can substitute AD with BC:
AB^2 = BC^2 + BD^2
4. Using Properties of Equilateral Triangle:
- Since AB = BC, we can rewrite the above equation as:
AB^2 = AB^2 + BD^2
BD^2 = 0 (as AB = AB)
5. Using the Given Information:
- Since AD = BC, we have:
4AD^2 = 4BC^2
3AB^2 = 4AD^2 (substituting BC with AD)
Therefore, the statement 3AB^2 = 4AD^2 is proved. Hence, option 'A' is correct.
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