Q12 of 56 Page 6

If ABC is an equilateral triangle of side 2a prove that the altitude AD = 3a and 3AB2 = 4AD2.

 
Given: Δ ABC is an equilateral triangle of side 2a. AD is the altitude of the triangle.
To Prove: AD = a√ 3 and 3AB2 = 4AD2
Proof:
In rt. Δ ADC,
AD2 = AC2 - DC2
    = (2a)2 - a2
    = 4a2 - a2
    = 3a2
∴ AD = a𢆣
3AB2 = 3(2a)2
     = 3(2a)2
     = 3 ´ 4a2
     = 4(a√ 3)2
     = 4AD2
∴ 3AB2 = 4AD2.

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