Question

In a triangle $ABC$, the co-ordinates of vertices $A,B$ and $C$ are $(4,7)$, $(-2,3)$ and $(0,1)$ respectively. find the equation of median through vertex $A$. Also, find the equation of the line through vertex $B$ and parallel to $AC$.

Answer 1

Midpoint $\left(D\right)$ of

$\begin{array}{c}BC=\left(\frac{-2+0}{2},\frac{3+1}{2}\right)\\ =\left(-1,2\right)\end{array}$

Equation of median through vertex $A$ is

$\begin{array}{c}y-7=\frac{2-7}{-1-4}\left(x-4\right)\\ \Rightarrow y-7=\frac{-5}{-58}\left(x-4\right)\\ \Rightarrow y-7=x-4\\ \Rightarrow x-y+3=0\end{array}$

Slope of line $AC=\frac{1-7}{0-4}=\frac{-6}{-4}=\frac{3}{2}$

Equation of line through vertex $B$ and parallel to $AC$ is

$\begin{array}{c}y+2=\frac{3}{2}\left(x+2\right)\\ \Rightarrow 2y+4=3x+6\\ \Rightarrow 3x-2y+2=0.\end{array}$