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State if the following is true or false.

In an equilateral triangle ABC, AD is the altitude drawn from A on side BC, then

3 $A B 2 = 4 A D 2 .$

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True

False

answer is A.

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Detailed Solution

Given that AD is the altitude drawn from A on side BC,
Question Image

The Pythagoras Theorem is given as,

H y p o t e n u s e 2 = B a s e 2 + P e r p e n d i c u l a r 2

Using the Pythagoras theorem,

A D 2 + B D 2 = A B 2

Substitute

12 B C

for BD in the equation

A D 2 + B D 2 = A B 2

to prove that

3 A B^{2} = 4 A D^{2}

⇒ A D 2 + 12 B C 2 = A B 2 ⇒ 4 A D 2 + A B 2 = 4 A B 2 ⇒ 4 A D 2 = 4 A B 2 − A B 2 ⇒ 4 A D 2 = 3 A B 2

The identity

3 A B 2 = 4 A D 2

has proved.
Hence, the statement is true.
Hence, option 1 is correct.

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