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Question

Question 15
In an equilateral triangle ABC, D is a point on side BC such that BD=13BC. Prove that 9AD2=7AB2.

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Solution


Let the side of an equilateral triangle be a, and AE be the altitude of ΔABC.
BE=EC=BC2=a2
And, AE=a32
given that, BD=13BC
BD=a3
DE=BEBD=a2a3=a6
Applying Pythagoras theorem in ΔADE, we get
AD2=AE2+DE2
AD2= (a32)2+(a6)2
AD2=(3a24)+(a236)
AD2=28a236
AD2=79AB2 [Since, the side of an equilateral triangle is a.i.e;AB=BC=AC=a]
9AD2=7AB2

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