Question

the equation of perpendicular bisector of line segment joining points A(4,5) & B(-2,3) is?
please answer with explanation and figure.​

Answer 1

Answer:

Hence, this is the answer.

Answer 2

Given:

Line segment AB having coordinates A(4,5) and B(-2,3)

To find:

Equation of perpendicular bisector of AB.

Solution:

A perpendicular line that passes through the midpoint of a line segment is called perpendicular bisector. Let midpoint of AB be M.

Midpoint formula = $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

where, $x_{1}=4,y_{1}=5,x_{2}=-2,y_{2}=3$

$M=(\frac{4+(-2)}{2},\frac{5+3}{2})$

$M=(\frac{2}{2},\frac{8}{2})$

$M=(1,4)$

The coordinates of the midpoint M of line segment AB is (1,4).

The slope of a line $(m)$ joining two points having coordinates $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Slope of line segment AB is given by $m_{1}$

$m_{1}=\frac{3-5}{-2-4}$

$m_{1}=\frac{-2}{-6} =\frac{2}{6}$

∴ $m_{1}=\frac{1}{3}$

Consider a point P above the line segment AB such that on joining the midpoint M to P, we get a straight line and PM is the perpendicular bisector of AB.

Let the slope of PM be $m_{2}$.

If two lines are perpendicular, then the product of their slopes is -1, i.e.,

$m_{1}*m_{2}=-1$

$\frac{1}{3}*m_{2}=-1$

$m_{2}=3*-1=-3$

∴ Slope of PM is calculated as $-3$.

The coordinates of P are unknown but coordinates of M have been determined and we also know the slope of PM. By point - slope formula, the equation of perpendicular bisector PM is given by:

$y-y_{1}=m(x-x_{1})$

where, $x_{1}=1,y_{1}=4$

$y-4=-3(x-1)$

$y-4=-3x+3$

$3x+y=3+4$

$3x+y=7$

$3x+y-7=0$

This is the equation of the perpendicular bisector of line segment AB.

The equation of the perpendicular bisector of line segment joining points A(4,5) and B(-2,3) is $3x+y-7=0$.