Question

Find the equation of the right bisector of the line segment joining the points $(3,4)$ and $(-1,2)$.

Answer 1

The right bisector of a line segment bisects the line segment at ${90}^{\circ }$.

The end-points of the line segment are given as $A(3,4)$ and $B(-1,2)$.

Accordingly, mid-point of $AB$ $=\{\frac{3-1}{2},\frac{4+2}{2}\}=(1,3)$
and slope of $AB$ $=\frac{2-4}{-1-3}=\frac{-2}{-4}=\frac{1}{2}$

$\therefore$ slope of line perpendicular to $AB$ $=-\frac{1}{\left\{\frac{1}{2}\right\}}=-2$

Thus equation of the line passing through $(1,3)$ and having a slope of $-2$ is given by,

$(y-3)=-2(x-1)$

$\Rightarrow y-3=-2x+2\Rightarrow 2x+y=5$
Thus, the required equation of the line is $2x+y=5$.