UPSC 2015 CDS I Math Paper

Remember: Median divides a triangle into two triangles of equal area. $\text{ar}(\triangle ABD)=\text{ar}(\triangle ACD)\ldots (1)$ (Given)Therefore, AD is the median of $\triangle ABC$ on base BC. O is any point on AD. So, OD is the median of $\triangle OBC$.$\therefore \text{ar}(\triangle OBD)=\text{ar}(\triangle OCD)\ldots (2)$Subtracting (2) from (1), we get$\text{ar}(\triangle ABD)-\text{ar}(\triangle OBD)=\text{ar}(\triangle ACD)-\text{ar}(\triangle OCD)$$\Rightarrow \text{ar}(\triangle ABO)=\text{ar}(\triangle ACO)$Hence, statement (1) is correct.G is the point of concurrence of the medians of $\triangle ABC$.$\text{ar}(\triangle ABG)=\text{ar}(\triangle ACG)\ldots (3)$(Proof same as given above)Also, $\text{ar}(\triangle BCG)=\text{ar}(\triangle ACG)\ldots (4)$From (3) and (4), we have$\text{ar}(\triangle ABG)=\text{ar}(\triangle BCG)=\text{ar}(\triangle ACG)$Hence, statement (2) is correct.