(Q12) Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.

Solution :

Given :∆ABC, an equilateral triangle;

AD - altitude and the side is a units, altitude h units.

R.T.P : 3a² = 4h²

Proof : In ∆ABD, ∆ACD

∠B = ∠ [∵ 60°]

∠ADB = ∠ [∵ 90°]

∴ ∠BAD = ∠ [∵ Angle sum property]

Also, BA = CA

∴ ∆ABD s ∆ACD (by SAS congruence condition)

Now in ∆ABD, AB² = ( )² + BD²

[∵ Pythagoras theorem]

⇒ 4h² = 3a²