Question

How do you find the equation of the perpendicular bisector of the points $(1,4)$and $(5, - 2)?$

Answer 1

Whenever they ask for an equation of perpendicular bisector, it is nothing but the point is at mid-point. So, first find the midpoint using midpoint formula given by: $midpo\operatorname{int} = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$ and then find the slope of two points using $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. Then by using the equation of line $y - {y_1} = m(x - {x_1})$ find the required equation.

Complete step by step solution:
In the given question they have asked to find the equation for perpendicular bisector where perpendicular bisector is a line which cuts the line exactly at the midpoint, which is shown as in the below diagram.

So now find the midpoint using midpoint formula given by:
$midpo\operatorname{int} = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$
Where ${x_1}$ and ${x_2}$ are coordinates of $x - axis$
${y_1}$ and ${y_2}$ are coordinates of $y - axis$
Here $({x_1},{y_1})$ is $(1,4)$ and $({x_2},{y_2})$ is $(5, - 2)$. Substituting these in the above midpoint formula, we get
$midpo\operatorname{int} = \left( {\dfrac{{1 + 5}}{2},\dfrac{{4 + ( - 2)}}{2}} \right)$
$\Rightarrow midpo\operatorname{int} = \left( {\dfrac{{1 + 5}}{2},\dfrac{{4 - 2}}{2}} \right)$
$\Rightarrow midpo\operatorname{int} = \left( {\dfrac{6}{2},\dfrac{2}{2}} \right)$
Therefore, $midpo\operatorname{int} = \left( {3,1} \right)$ .
Now, to find the equation of a line, we need to find slope by using the formula given by:
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ where $m$ is slope
By substituting the values into the slope equation we get
$m = \dfrac{{ - 2 - 4}}{{5 - 1}}$
$\Rightarrow m = \dfrac{{ - 6}}{4}$
$\Rightarrow m = \dfrac{{ - 3}}{2}$
Given a line with slope $m$ then the slope of a line perpendicular to it can be calculated as
${m_{perpendicular}} = - \dfrac{1}{m}$
On substituting the value of slope, we get
$\Rightarrow {m_{perpendicular}} = - \dfrac{1}{{ - \dfrac{3}{2}}}$
$\Rightarrow {m_{perpendicular}} = \dfrac{2}{3}$
Now, to find the equation of line we have a formula
$y - {y_1} = m(x - {x_1})$
Here, ${x_1}$ and ${y_1}$ is mid-point which is $\left( {3,1} \right)$
Therefore, equation of line becomes,
$y - 1 = \dfrac{2}{3}(x - 3)$
$\Rightarrow y - 1 = \dfrac{2}{3}x - 2$
$\Rightarrow y = \dfrac{2}{3}x - 2 + 1$

$\Rightarrow y = \dfrac{2}{3}x - 1$ is the required equation.

Note:
Whenever they ask to find the equation of two points first try to understand whether the given problem is on mid-point or not and find the slope of the line and finally solve for the equation. Once you find the slope if it is asked for a perpendicular bisector form then find slope for that or else the answer you get is not the correct one.