$$D$$ is a point on $$AC$$  of the triangle with vertices $$A\left( {2,\,3} \right),\,B\left( {1,\, - 3} \right),\,C\left( { - 4,\, - 7} \right)$$       and $$BD$$  divides $$ABC$$   into two triangles of equal area. The equation of the line drawn through $$B$$ at right angles to $$BD$$  is :

Solution :
Since, the line $$BD$$  divides the triangle into two of equal area. $$BD$$  is a median and $$D$$ is $$\left( { - 1,\, - 2} \right).$$
Slope of $$BD = - \frac{1}{2}$$

So, the required line is
$$\eqalign{ & y + 3 = 2\left( {x - 1} \right) \cr & \Rightarrow y - 2x + 5 = 0 \cr} $$