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$A B C$ 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
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The correct answer is:
$\left(\frac{19}{2}, 6\right)$
We know that, centroid divides median in the ratio 2:1.
$\begin{aligned}(7,5) &=\left(\frac{2 x+2}{2+1}, \frac{2 y+3}{2+1}\right) \\(7,5) &=\left(\frac{2 x+2}{3}, \frac{2 y+3}{3}\right) \\ \therefore \quad \frac{2 x+2}{3} &=7 \text { and } \frac{2 y+3}{3}=5 \\ \Rightarrow \quad x &=\frac{19}{2} \text { and } y=6 \\ \therefore D & \text { is }\left(\frac{19}{2}, 6\right) . \end{aligned}$
$\begin{aligned}(7,5) &=\left(\frac{2 x+2}{2+1}, \frac{2 y+3}{2+1}\right) \\(7,5) &=\left(\frac{2 x+2}{3}, \frac{2 y+3}{3}\right) \\ \therefore \quad \frac{2 x+2}{3} &=7 \text { and } \frac{2 y+3}{3}=5 \\ \Rightarrow \quad x &=\frac{19}{2} \text { and } y=6 \\ \therefore D & \text { is }\left(\frac{19}{2}, 6\right) . \end{aligned}$
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