Q. 154.3( 250 Votes )

In an equilateral triangle ABC, D is a point on side BC such that Prove that 9 AD= 7 AB2

Answer :

The figure is given below:

Given: BD = BC/3

To Prove: 9 AD2 = 7 AB2

Let the side of the equilateral triangle be a, and AM be the altitude of ΔABC

BM = MC = BC/2 = a/2                              [ Altitude of an equilateral triangle bisect the side]

And, then, in ΔABM, by pythagoras theorem we write,
Pythagoras Theorem : Square of the Hypotenuse equals to the sum of the squares of other two sides.


or AM= a- a2/4 

BD = a/3                                       [ BC = a]

DM = BM – BD

= a/2 – a/3

= a/6

According to pythagoras theorem in a right angled triangle,
(hypotenuse)= (altitude)2 + (base)2

Applying Pythagoras theorem in ΔADM, we obtain

AD2 = AM2 + DM



Now, a = AB or a = AB2

 36 AD= 28 AB2

  9 AD2 = 7 AB
Hence, Proved

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