The correct option is
A 4(BL2+CM2)=5BC2Given:
ΔABC right angled at
A, i.e.
∠A+900 where
BL and
CM are the medians
Proof:- Since BL is median,
AL=CL=12AC...........(1)
Similarly, CM is the median
AM=MB=12AB...........(2)
We know that, by Pythagoras theorem
(hypotimise)2=(height)2+(Base)2
In ΔBAC,
(BC)2=(AB)2+(AC).........(1)
In ΔBAL,
(BL)2=(AB)2+(AL).........(1) From(1):AL=12AC
=(AB)2+(AC2)2
=(AB)2+(AC)24)2
(BL)2=4(AB)2+(AC)24
4BL2=4(AB)2+AC2,,,,,,,,,(ii)
In ΔMAC
(CM)2=(AM)2+(AC)2 From(2)AM=12AB
(CM)2=(AB2)2+(AC)2
(CM)2=(AB)222+(AC)2
CM2=(AB)2+4(AC)24
4(CM)2=(AB)2+4(AC)2............(iii)
adding (ii) and (iii)
4(BL)2+4(CM)2=4(AB)2+(AC)2+(AB)2+4(AC)2
4(BL)2+(CM)2=5(AB)2+5(AC)2
4(BL)2+(CM)2=5(AB)2+5(AC)2
4(BL)2+(CM)2=5(BC)2
From(BC)2=(AB)2+(AC)2
Hence, proved