Question

IN A EQUILATERAL TRIANGLE ABC AD IS PERPENDICULAR TO BC.PROVE THAT AB^2=4AD^2

Answer 1

ABC IS EQUILATERAL THEREFORE AB = BC =CA

AD IS PERPENDICULAR TO BC THEREFORE BD = 1/2AB, DC = 1/2 AB

IN TRIANGLE ABD AB2 = AD2 +BD2 -----------------1

IN TRIANGLE ACD AC2 = AD2 +DC2 ----------------2

1 + 2 GIVES AB2 +AC2 = 2AD2 + BD2 + DC2

AB2 + AB2 = 2AD2 +1/4AB2 + 1/4 AB2

2AB2 = 2AD2 +1/2 AB2

(2-1/2)AB2 = 2AD2

3/2AB2 = 2AD2

3AB2 = 4AD2