Question

in an equilateral triangle ABC,AD perpendicular to BE THEN prove AD^2=3BD^2

Answer 1

Note that ABDABD is a right triangle, thus:

AB2=AD2+BD2AB2=AD2+BD2

You have also BC=AB=2BDBC=AB=2BD

Then 4BD2=AD2+BD24BD2=AD2+BD2

That is

AD2=3BD2

Answer 2

Hi !

In an equilateral triangle ABC ,
AB = BC = AC
AD ⟂ BC
AD is also the perpendicular bisector of the side BC.
Hence, BD = DC
Also,
BC = 2 BD

According to Pythagoras theorem,
In ΔADB,

AB² = AD² + BD²

AD² = AB² - BD²

AB = BC ,

AD² = BC² - BD²

AD² = (2BD)² - BD²

AD² = 4BD² - BD²

AD² = 3BD²

Proved.