QUESTION

# Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3, 5).

Hint: Perpendicular bisector is a perpendicular line which passes through the midpoint of the line joining the given two points. Start with finding the midpoint of the given points and then use this point to find the equation of line perpendicular to the line formed by the given two points.

Let the given points (7, 1) and (3, 5) be A and B respectively.
Also suppose point M is the midpoint of points A (7, 1) and B (3, 5). And we know that the coordinates of midpoint of two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$.
So the coordinates of points M are:
$\therefore$Midpoint of $\left( {AB} \right) = \left( {\dfrac{{7 + 3}}{2},\dfrac{{5 + 1}}{2}} \right) = \left( {\dfrac{{10}}{2},\dfrac{6}{2}} \right) = M\left( {5,3} \right)$
Further, we know that the slope of line joining two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.
Therefore, the slope of the line joining points A and B is ${m_1} = \dfrac{{3 - 7}}{{5 - 1}} = \dfrac{{ - 4}}{4} = - 1$
Perpendicular bisector of points A and B will be the line perpendicular to AB. Since the product of slopes of two perpendicular lines is -1, we have:
$\Rightarrow {m_1} \times {m_2} = - 1$, where ${m_2}$ is the slope of the perpendicular bisector.
$\Rightarrow - 1 \times {m_2} = - 1$
$\Rightarrow {m_2} = 1$
Perpendicular bisector will pass through the points A and B i.e. point M.
In this case, the perpendicular bisector is eventually a line passing through point $M\left( {5,3} \right)$ and having slope ${m_2} = 1$.
According to point slope form of equation of line, if a line of slope $m$ is passing through a point $\left( {{x_1},{y_1}} \right)$, its equation is:
$\Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
So, using point slope form of the equation of line, the equation of perpendicular bisector is:
$\Rightarrow (y - 3) = 1(x - 5)$
$\Rightarrow y - 3 = x - 5$
$\Rightarrow x - y - 2 = 0$
Thus the equation of the perpendicular bisector is $x - y - 2 = 0$.

Note: In the above problem, we have used point slope form to find the equation of line.
If the line is passing through two known points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$, then its equation is:
$\Rightarrow \left( {y - {y_1}} \right) = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\left( {x - {x_1}} \right)$
There are various other ways to find the equation of line. We can use different methods as per the data available to us.