in triangle BAC ,angle c=90 and q is mid point of bc.prove that AB square =4AQ square-3AC square

Answer : Given : in triangle bac ,angle c=90 and q is mid point of bc.

To prove :ab²  =4aq² - 3ac²

Now let us take RHS of the proving statement, we get

= 4aq² - 3ac²

= 4(qc²+ac²)- 3ac^{2  {using pythagorus theorm in triangle qac}}

= 4qc²+ac² 

= 4 (bc/2)² +ac² { since q is mid point of side bc ,qc= bc/2}

= bc² + ac²

= ab² {using pythagorus theorm in triangle bac } 

= LHS

hence proved