question_answer

If in \[\Delta ABC,\]\[A=\,(1,10)\], circumcentre \[\left( -\frac{1}{3},\frac{2}{3} \right)\]and orthocentre \[\left( \frac{11}{3},\frac{4}{3} \right)\] The coordinates of the mid-point  of the sides opposite to A is

A) \[(1,5)\]

B) \[\left( 1,-\frac{11}{3} \right)\]

C) \[(1,-\,3)\]

D) \[(1,6)\]

Correct Answer: B

Solution :

We know that centroid divides the orthocentre and circumcentre in the ratio \[2:1\].

\[G=\left( \frac{-\frac{2}{3}+\frac{11}{3}}{3},\frac{\frac{4}{3}+\frac{4}{3}}{3} \right)\]

\[G=\left( 1,\frac{8}{9} \right)\]

Also, \[AG:GD=2:1\]

\[1=\frac{2h+1}{3},\frac{8}{9}=\frac{2k+10}{3}\]

\[h=1,\]\[k=-\frac{11}{3}\]

\[\therefore \]Coordinate of mid-point of opposite side of A is \[\left( 1,-\frac{11}{3} \right)\]