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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 it is equilateral.
Consider ABC,
ADB = 90
ABC = 60
BAD=1806090=30
Similarly, CAD=30
Now, ABDACD
According to SAS congruency,
Therefore, BD = DC = a2.
From Pythagoras' theorem,
AD2+BD2=AB2
AD=a2(a2)2
AD=3a2
AD×cotθ=AD×cot 30=AD×ADBD=3a2×3a2a2
AD×cot θ=3a2


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