Question

if in aequilatarel triangleABC,AD PERPENDICULAR BC, then prove that AD square=3BDsquare

Answer 1

in an equilateral triangle the median,altitude and the angle bisector coincide with each other; so AD is the perpendicular bisector of BC;
let AB=BC=CA=a;
⇒ BD = a/2
⇒BD^2 = (a^2)/4                    ...............(1)
in right triangle ABD;
AB^2 = AD^2 + BD^2
⇒AD^2 = AB^2 - BD^2
⇒AD^2 = a^2 - (a/2)^2
⇒AD^2 = 3(a^2)/4                  ................(2)
from (1) & (2);
AD^2 = 3(BD^2)