Q. 113.9( 43 Votes )

In <span lang="EN

Answer :

We know, By Apollonius theorem


In ΔABC, if L is the midpoint of side AC, then AB2 + BC2 = 2BL2 + 2AL 2


Given that, BL is median i.e. L is the mid-point of CA



AB2 + BC2 = 2BL2 + 2AL2



[1]


Also, if M is the midpoint of side AB, then AC2 + BC2 =2CM2 + 2BM2


Given that, CM is median i.e. M is the mid-point of BA



AC2 + BC2 = 2CM2 + 2BM2



[2]


Also, In ΔABC, By Pythagoras theorem i.e.


(Hypotenuse)2 = (base)2 + (Perpendicular)2


BC2 = AC2 + AB2 [3]


Adding [1] and [2]




AB2 + AC2 + 4BC2 = 4(BL2 + CM2)


BC2 + 4BC2 = 4(BL2+ CM2) [From 3]


5BC2 = 4(BL2+ CM2)


Hence Proved.


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