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Question

In an equilateral triangle ABC of side 'a', AD is the perpendicular drawn onto side BC. Find AD cot θ in terms of a.

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Solution

In ABC,

A=B=C=60
(since ABC is equilateral)

Since altitude of an equilateral triangle bisects the angle of vertex.
BAD=30

Since altitude of an equilateral triangle bisects the side.
Therefore, BD = DC = a2.

From Pythagoras theorem,
AD2+BD2=AB2
AD=a2(a2)2
AD=3a2
Now, AD cotθ=AD cot 30=AD×3
AD cot θ=3a2×3
AD cot θ=3a2

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