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Question

ABC is a right angled triangle at C. If D is the mid point of BC,prove that AB2=4AD2-3AC2


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
Hence proved

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