question_answer

Two medians AD and BE of\[\Delta ABC\] intersect at G at right angles. If \[AD=9\,\,cm\] and \[BE=9\,\,cm\]then the length of BD, in cm, is                                                                                                  [SSC (CPO) 2012]

A) 10                                

B) 6     

C) 5         

D) 3

Correct Answer: C

Solution :

\[AD=9\,\,cm\]

\[\because \]A centroid divides the median in the ratio 2: 1.

\[\therefore \]      \[GD=\frac{1}{3}\times 9=3\,\,cm\]

[\[\because \]point of intersection of median]

and \[BG=\frac{2}{3}\times BE\]\[\Rightarrow \]\[BG=\frac{2}{3}\times 6=4\,\,cm\]

Now, in \[\Delta BDG\] by Pythagoras theorem,

\[B{{D}^{2}}=B{{G}^{2}}+G{{D}^{2}}\]

\[\therefore \]      \[BD=\sqrt{{{3}^{2}}+{{4}^{2}}}=\sqrt{9+16}\]

\[\Rightarrow \]   \[BD=\sqrt{25}=5\,\,cm\]