Question

Consider triangle ABC, Prove that : AC² = AM² + 3BM²​

Answer 1

hope it helps... thanks

Answer 2

Given :

• ABC is a right angled triangle

• CM = BM

To prove :

• ( AC )² = ( AM )² + ( 3BM )²

Proof :

In ∆ ABC :-

➨ BC = 2BM

By Pythagoras Theorem

➨ ( AC )² = ( AB )² + ( BC )²

➨ ( AB )² = ( BC )² - ( AC )² ----(1)

In ∆ ABM :-

➨ ( AM )² = ( AB )² + ( BM )²  

➨ ( AB )² = ( BM )² - ( AM )²  ----(2)

FROM (1) & (2)

➨ ( BC )² - ( AC )² = ( BM )² - ( AM )²

➨ ( BC )²- ( BM )²+ ( AM )² = ( AC )²

➨ ( 2BM )² - ( BM )² + ( AM )² = ( AC )²    

➨ ( 4BM )² - ( BM )² + ( AM )² = ( AC )²

➨ ( 3BM )² + ( AM )² = ( AC )²

➨ ( AC )² = ( AM )² + ( 3BM )²

HENCE PROVED