Question

In an equilateral triangle ABC,AD is perpendicular toBC, then find 3ABsquar =?

Answer 1

ABC is an equilateral triangle therefore AB=BC=AC
AD is perpendicular to BC, therefore BD=DC=1/2BC=1/2AB
In ΔABC, AB²=AD²+BD²   .....(1)
In ΔACD, AC²=AD²+DC²  .......(2)
(1)+(2)
AB²+AC²=AD²+BD²+AD²+DC²
            =2AD²+BD²+DC²
AB²+AB²=2AD²+1/4AB²+1/4AB²            
2AB²=2AD²+1/2AB²
(2-1/2AB²)=2AD²
3/2AB²=2AD²
3AB²=4AD²

Hope it helps.
            

Answer 2

LOOK AT THE PIC GIVEN BELOW

AB=BC=AC
LOOK INTO ΔABD
ANGLES ARE 30,60,90
SIDES RATIO=X:X√3:2X
AB=2X
AB²=4X²
AND AD=X√3
AD²=3X²

3AB²=4X²*3=12X²
SO WE CAN SAY THAT
AD²*4=12X²
(3X²*4)=12X²

WE PROVED THAT 4AD²=3AB²