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The vertices of $\Delta PQR$ are $P (2,1), Q (-2,3)$ and $R (4,5) .$ Find equation of the median through the vertex $R$.
$3 x-4 y+8=0$
$3 x-4 y+8=0$
$3 x-4 y+8=0$
$3 x-4 y+8=0$
Solution
It is given that the vertices of $\Delta PQR$ are $P (2,1), Q (-2,3)$ and $R (4,5)$.
Let $RL$ be the median through vertex $R$.
Accordingly, $L$ be the mid-point of $PQ$.
By mid-point formula, the coordinates of point $L$ are given by $\left(\frac{2-2}{2}, \frac{1+3}{2}\right)=(0,2)$
It is known that the equation of the line passing through points
$\left(x_{1}, y_{1}\right)=(4,5)$ and $\left(x_{2}, y_{2}\right)=(0,2).$
Hence, $y-5=\frac{2-5}{0-4}(x-4)$
$\Rightarrow y-5=\frac{-3}{-4}(x-4)$
$\Rightarrow 4(y-5)=3(x-4)$
$\Rightarrow 4 y-20=3 x-12$
$\Rightarrow 3 x-4 y+8=0$
Thus, the required equation of the median through vertex $R$ is $3 x-4 y+8=0$.
