Question

The vertices of $\Delta PQR$ are $P(2,1),Q(-2,3)$ and $R(4,5)$. Find equation of the median through the vertex $R$.

Answer 1

Step $1:$ Simplification given data

The vertices of $\Delta PQR$ are $P(2,1),Q(-2,3)$ and $R(4,5)$
We need to find equation of median i.e.
Equation of $RS$
Since $RS$ is median, $S$ is the midpoint of $PQ.$
We know that midpoint of a line joining points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
Mid-point of $PQ$ joining the points $P(2,1)$and $Q(-2,3)$ is
$S=(\frac{2+(-2)}{2},\frac{1+3}{2})$
$S=(\frac{2-2}{2},\frac{4}{2}),S=(0,2)$

Step $2:$ Equation of median
We know that equation of line through two points $(x_1,y_1)$ and $(x_2,y_2)$ is
$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$
Equation of line passing through $R(4,5)$ and $S(0,2)$
$\Rightarrow (y-5)=\frac{2-5}{0-4}(x-4)$
$\Rightarrow y-5=\frac{-3}{-4}(x-4)$
$\Rightarrow y-5=\frac{3}{4}(x-4)$
$\Rightarrow 4(y-5)=3(x-4)$
$\Rightarrow 4y-20=3x-12$
$\Rightarrow 4y-3x-20+12=0$
$\Rightarrow 4y-3x-8=0$
$\Rightarrow 3x-4y+8=0$
Therefore, equation of median is $3x-4y+8=0$.