ABC is a right angled triangle at C. If D is the mid point of BC,prove that AB2=4AD2-3AC2
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Solution
GIVEN:- ΔABC is right angled at C. D is the mid-point of BC. T.P. :- AB2 = 4AD2 - 3AC2 Proof :- Since D is the mid point of BC DC=BD and ... DC=BC/2 ............(i) In ΔABC BY P.G.T(Pythagores theorm) AB2 = AC2 + BC2 .........(ii) IN ΔACD BY P.G.T (AD)2 = (AC)2 + (DC)2 ............(iii) Putting (i) in (iii) (AD)2 = (AC)2 + (BC / 2)2 AD2 = AC2 + (BC)2 4 AD2 = 4AC2 + (BC)2 4 4AD2 = 4AC2 + BC2 4AD2 = 3AC2 + AC2+ BC2
4AD2 - 3AC2 = AC2+ BC2 ........(iv) Putting (ii) in (iv) 4AD2 - 3AC2 = AB2