$A B C$ is a triangle, $G$ is the centroid, $D$ is the mid-point of $B C$. If $A=(2,3)$ and $G=(7,5)$, then the point $D$ is

• Correct answer
  A

  $\left(\frac{19}{2}, 6\right)$
• B

  $\left(\frac{9}{2}, 4\right)$
• C

  $\left(8, \frac{13}{2}\right)$
• D

  $\left(\frac{11}{2}, \frac{11}{2}\right)$

Solution

We know that, centroid divides median in the ratio 2:1.

$\begin{aligned}(7,5) &=\left(\frac{2 x+2}{2+1}, \frac{2 y+3}{2+1}\right) \\(7,5) &=\left(\frac{2 x+2}{3}, \frac{2 y+3}{3}\right) \\ \therefore \quad \frac{2 x+2}{3} &=7 \text { and } \frac{2 y+3}{3}=5 \\ \Rightarrow \quad x &=\frac{19}{2} \text { and } y=6 \\ \therefore D & \text { is }\left(\frac{19}{2}, 6\right) . \end{aligned}$

Hence, the correct answer is option: 'A'.