Question

In an equilateral triangle ABC, AD is perpendicular to BC , prove AD²=3BD²

Answer 1

Here is your answer
Here given triangle ABC must be an equilateral triangle and we have to prove AD2 =  3BD

In right ∆ABD,

AB2 = BD2 + AD2

⇒ BC2 = BD2 + AD2   (AB = BC)

⇒ (2BD)2 = BD2 + AD2   [From (1)]

⇒ 4BD2 = BD2 + AD2

⇒ AD2 = 3BD2 = 3DC2  [As BD = DC]

Answer 2

Ad perpendicular bc
BD=DC=1/2BC
2bd=bc
In right ∆ABD,
AB2 = BD2 + AD2
⇒ BC2 = BD2 + AD2 (AB = BC)
⇒ (2BD)2 = BD2 + AD2 [From (1)]
⇒ 4BD2 = BD2 + AD2
⇒ AD2 = 3BD2 = 3DC2 [As BD = DC]
Hence proved