Question

In $∆ABC$, if $A\left(3,5\right)$, $B\left(7,8\right)$ and $C\left(1,-10\right)$, then find the equation of the median through $A$.

Answer 1

Step 1: Finding the midpoint of side $BC$:
We are given that: $B\left(7,8\right)$ and $C\left(1,-10\right)$
Let $O\left(x,y\right)$ be the midpoint of side $BC$.
Using midpoint formula, we get:
$x=\frac{x_1+x_2}{2}$ and $y=\frac{y_1+y_2}{2}$
$\Rightarrow x=\frac{7+1}{2}$and $y=\frac{8-10}{2}$
$\Rightarrow x=4$ and $y=-1$
$\Rightarrow O(x,y)=\left(4,-1\right)$

Step 2: Finding the equation of the median through $A$:
We know that, the median through $A$ will bisect side $BC$ at $O$.
Also, we are given that: $A\left(3,5\right)$
Equation of a line in two point form is:
$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$
Equation of the median through $A$is given as:
$y+1=\frac{5+1}{3-4}\left(x-4\right)$
$\Rightarrow y+1=-6x+24$
$\Rightarrow 6x+y-23=0$

Hence, the equation of the median through $A$ is $6x+y-23=0$.