Question : If ABC is an equilateral triangle and D is a point in BC such that AD is perpendicular to BC, then:
Option 1: AB : BD = 1 : 1
Option 2: AB : BD = 1 : 2
Option 3: AB : BD = 2 : 1
Option 4: AB : BD = 3 : 2
Correct Answer: AB : BD = 2 : 1
Solution :
In $\triangle$ABC, AD$\perp$BC
As $\triangle$ABC is an equilateral triangle, AD is an altitude as well as median on BC, i.e., AD is a perpendicular bisector of BC.
⇒ BD = DC = $\frac{1}{2}$BC
Since AB = BC = CA,
⇒ BD = $\frac{1}{2}$BC = $\frac{1}{2}$AB
⇒ $\frac{AB}{BD}$ = $\frac{2}{1}$
⇒ AB : BD = 2 : 1
Hence, the correct answer is AB : BD = 2 : 1.
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