Question

BL and CM are medians of a ∆ABC right angled at A. Prove that 4( BL

2 + CM2

) = 5 BC

2​

Answer 1

Answer:

QUESTION

BL and CM are medians of a ∆ABC right angled at A. Prove that 4( BL

2 + CM2

) = 5 BC

2

ANSWER

BL is median

BL is median⇒ AL=CL=1/2. AC  eq.(1)

(1)CM is median

(1)CM is median⇒ AM = MB = 1/2 AB  eq.(2)

(2)InΔBAC

(2)InΔBAC(BC)^2=(AB)^2+(AC)^2

InΔBAC

InΔBAC(BL)^2=(AB)^2+(AC/2)^2

24BL^2=4AB^2+(AC)^2

2In ΔMAC,

2In ΔMAC,(CM)^2=(AM)^2+(AC)^2

2(CM)^2=(AB/2)^2+(AC)^2

24CM^2=(AB)^2+(AC)^2

NOW,(BC)^2=(AB)^2+(AC)^2  ( eq.(1))

4BC2=4(AB)2+(AC)2  ( eq(2))

4CM2=AB2+4AC2  (eq(3))

ADD  Eq.(2)&(3)

(2)&(3) 4BC^2+4CM^2=5AB^2+5AC^2

24(BL^2+CM^2)=5BC^2

Hence proof