Question

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

Answer 1

Hint: We know that the right bisector of a line segment is passing through its midpoint and perpendicular to it. We will find the midpoint using the formula as follows:
If we have end points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$.
X coordinate of midpoint $=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$
Y coordinate of midpoint $=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$
Also we will use the property that the product of the slope of a line and a line perpendicular to it is equal to minus one.

Complete step-by-step answer:
We have been asked to find the right bisector of the line segment the points (3, 4) and (-1, 2).
We know that the right bisector of the line segment is perpendicular and passing through the midpoint of the line segment.
We know the midpoint of line segment joining points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by:
X coordinate of midpoint $=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$
Y coordinate of midpoint $=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$
So midpoint of the line segment joining the points (3, 4) and (-1, 2) is given by:
X coordinate of midpoint $=\dfrac{3-1}{2}=\dfrac{2}{2}=1$
Y coordinate of midpoint $=\dfrac{4+2}{2}=\dfrac{6}{2}=3$
Hence the coordinate of the point is (1,3).
Also, we know that if we have two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ then slope is given by:
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
So the slope of line segment joining (3, 4) and (-1, 2) is given by:
$m=\dfrac{2-4}{-1-3}=\dfrac{-2}{-4}=\dfrac{1}{2}$
Since we know that the product of a line and its perpendicular line is equal to minus one
Let the slope of right bisector be ${{m}_{1}}$

& \Rightarrow m{{m}_{1}}=-1 \\\ & \Rightarrow \dfrac{1}{2}\times {{m}_{1}}=-1 \\\ & \Rightarrow {{m}_{1}}=-2 \\\ \end{aligned}$$ We know the equation of a line in slope and point form is given by: $$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$$ where $$\left( {{x}_{1}},{{y}_{1}} \right)$$ is a point through which the line passes and m is the slope. So the equation of right bisector passing through (1,3) and having slope -2 is given by: $$\begin{aligned} & y-3=-2\left( x-1 \right) \\\ & y-3=-2x+2 \\\ & y+2x-3-2=0 \\\ & y+2x-5=0 \\\ \end{aligned}$$ Therefore, the required equation of the right bisector of the line segment is $$y+2x-5=0$$. Note: Be careful while finding the values of midpoint and the slope of the line segment as there is a chance of sign mistake during calculation. It must be remembered that while finding the midpoint, we add the coordinates and not subtract them. This is probably a common silly mistake that can be made. Also, remember the property that if two lines are perpendicular then the product of their slope is equal to minus one.