question_answer

If AD, BE and CF are the medians of \[\Delta \,ABC\] then which one of the following statements is correct?                                 

A)  (AD 4- BE + CF) = (AB + BC + CD)       

B)  (AD + BE + CD) > -3 (AB + BC + CA)       

C)  (AD + BE + CF) < 3- (AB + BC + CA)     

D)  (AD + BE + CF) = - (AB + BC + CA)

Correct Answer: B

Solution :

 From \[\Delta \,BCG,\] \[BG+GC>BC\]                 or            \[\frac{2}{3}BE+\frac{2}{3}CF>BC\]                 or            \[BE+CF>\frac{3}{2}BC\]               ??.(i) Similarly,                 \[CF+AD>\frac{3}{2}CA\]              ??.(ii) and        \[AD+BE>\frac{3}{2}AB\]              ??(iii)                 Adding equations (i), (ii) and (Hi), we get \[2(AD+BE+CF)>\frac{3}{2}(AB+BC+CA)\] or            \[AD+BE+CF>\frac{3}{4}(AB+BC+CA)\]