Question : If $AD, BE$ and $CF$ are medians of $\triangle ABC$, then which of the following statement is correct?
Option 1: $(AD + BE + CF) > (AB + BC + CA)$
Option 2: $(AD + BE + CF) < (AB + BC + CA)$
Option 3: $(AD + BE + CF ) = (AB + BC + CA)$
Option 4: $(AD + BE + CF ) = \sqrt2(AB+BC+CA)$
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Correct Answer: $(AD + BE + CF) < (AB + BC + CA)$
Solution :
Here, $AB+AC>2AD$ ---------------(1)
$AB+BC>2BE$ -------------------(2)
$BC+AC>2CF$ -------------------(3)
By adding equation (1), (2), and (3), we get,
$2(AB+BC+AC)>2(AD+BE+CF)$
$⇒(AB+BC+AC)>(AD+BE+CF)$
Hence, the correct answer is $(AD + BE + CF) < (AB + BC + CA)$.
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