In an equilateral △ABC, let AD ⟂ BC. Which relation between AB and AD is true?
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A3AB2 = 2AD2
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B2AB2 = 3AD2
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C3AB2 = 4AD2
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D4AB2 = 3AD2
Answer
Correct Answer: 3AB2 = 4AD2
Explanation
Introduction / Context:In an equilateral triangle of side s, the altitude AD has a standard length related to s. Substituting that value yields identities among AB and AD.
Given Data / Assumptions:
- AB = s.
- Altitude AD = (√3/2)s.
Concept / Approach:Compute AB² and AD², then test which given identity holds exactly without approximation.
Step-by-Step Solution:
AB² = s²AD² = (3/4)s²Check 3AB² = 3s² and 4AD² = 4*(3/4)s² = 3s² ⇒ 3AB² = 4AD² (true)Verification / Alternative check:Other listed relations evaluate to unequal expressions when substituting AB² and AD² as above.
Why Other Options Are Wrong:They swap coefficients incorrectly; only the 3:4 relation matches altitude length in an equilateral triangle.
Common Pitfalls:Using median length s/2 or height s*(√3/3) (which is the inradius) instead of the altitude length (√3/2)s.
Final Answer:3AB2 = 4AD2