Question

What is the equation of the line that is perpendicular to the line passing through `(-5,3)` and `(4,9)` at midpoint of the two points?

Answer 1

`y=-1 1/2x+2 1/4`

Explanation:

The slope a line that is perpendicular to a given line would be the inverse slope of the given line

`m = a/b` the perpendicular slope would be `m =-b/a`

The formula for the slope of a line based upon two coordinate points is

`m = (y_2-y_1)/(x_2-x_1)`

For the coordinate points `(-5,3) and (4,9)`
`x_1 = -5`
`x_2 = 4`
`y_1 = 3`
`y_2 = 9`

`m = (9-3)/(4-(-5))`

`m = 6/9`

The slope is `m = 6/9`
the perpendicular slope would be the reciprocal (-1/m)
`m = -9/6`

To find the midpoint of the line we must use the midpoint formula

`((x_1+x_2)/2,(y_1+y_2)/2)`

`((-5+4)/2,(3+9)/2)`

`(-1/2,12/2)`

`(-1/2,6)`

To determine the equation of the line use the point slope form
`(y-y_1)=m(x-x_1)`

Plug in the midpoint in order to find the new equation.
`(-1/2,6)`

`(y-6)=-9/6(x-(-1/2))`

`y-6=-9/6x-9/12`

`ycancel(-6)cancel(+6)=-1 1/2x-3/4 +3`

`y=-1 1/2x+2 1/4`