Question

write an equation in point-slope form for the perpendicular bisector of the segment with endpoints at A(-2,2) and B(5,4)

Answer 1

The equation in point-slope form for the perpendicular bisector of the segment with endpoints at A(-2,2) and B(5,4) is
`y - 3 = (-7x)/(2)+ (21)/(4)`

Solution:

Given that we have to write equation in point-slope form for the perpendicular bisector of the segment with endpoints at A(-2,2) and B(5,4)

Let us first find the slope of given line AB

The slope "m" of the line is given as:

`m=(y_(2)-y_(1))/(x_(2)-x_(1))`

Here the given points are A(-2,2) and B(5,4)

`\text {Here } x_(1)=-2 ; y_(1)=2 ; x_(2)=5 ; y_(2)=4`

`m=(4-2)/(5-(-2))=(2)/(7)`

Thus the slope of line with given points is
`(2)/(7)`

We know that product of slopes of given line and slope of line perpendicular to given line is always -1

`\begin{array}{l}{\text {slope of given line } * \text { slope of perpendicular bisector }=-1} \\\\ {(2)/(7) * \text { slope of perpendicular bisector }=-1} \\ \\{\text {slope of perpendicular bisector }=(-7)/(2)}\end{array}`

The perpendicular bisector will run through the midpoint of the given points

So let us find the midpoint of A(-2,2) and B(5,4)

The midpoint formula for given two points is given as:

`\text {For two points }\left(x_(1), y_(1)\right) \text { and }\left(x_(2), y_(2)\right), \text { midpoint } \mathrm{m}(x, y) \text { is given as }`

`m(x, y)=\left((x_(1)+x_(2))/(2), (y_(1)+y_(2))/(2)\right)`

Substituting the given points A(-2,2) and B(5,4)

`m(x, y)=\left((-2+5)/(2), (2+4)/(2)\right)=\left((3)/(2), 3\right)`

Now let us find the equation of perpendicular bisector in point slope form

The perpendicular bisector passes through points (3/2, 3) and slope -7/2

The point slope form is given as:

`y - y_1 = m(x - x_1)`

`\text { Substitute } \mathrm{m}=(-7)/(2) \text { and }\left(x_(1), y_(1)\right)=\left((3)/(2), 3\right)`

`y - 3 = (-7)/(2)(x - (3)/(2))\\\\y - 3 = (-7x)/(2)+ (21)/(4)`

Thus the equation in point-slope form for the perpendicular bisector of the segment with endpoints at A(-2,2) and B(5,4) is found out