question_answer

AX, BY and CZ are the medians of \[\Delta \,ABC\] intersecting at O. If CK is drawn parallel to BY to meet AX in K, then AO is

A) \[\frac{1}{4}\,AK\]

B) \[\frac{1}{2}\,AK\]

C) \[\frac{1}{3}\,AK\]

D) \[\frac{2}{3}\,AK\]

Correct Answer: B

Solution :

   Since angles BXO and CXK are similar.                 \[\therefore \]                  \[\frac{OX}{XK}=\frac{BX}{OX}=1\]                 \[\Rightarrow \]               \[OX=XK\] Also>      \[\frac{AO}{XK}=\frac{AO}{OX}=\frac{2}{1}\] \[\Rightarrow \]               \[AO=OK\] Hence,   \[AK=2AO\] \[\Rightarrow \]               \[AO=\frac{1}{2}AK\]