Question

What is the equation of the line that is perpendicular to the line passing through `(-5,12)` and `(4,-3)` at midpoint of the two points?

Answer 1

See a solution process below:

Explanation:

First, we need to find the mid-point and of the two points and the slope of the line going through the two points.

The formula to find the mid-point of a line segment give the two end points is:

`M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)`

Where `M` is the midpoint and the given points are:

`(color(red)(x_1), color(red)(y_1))` and `(color(blue)(x_2), color(blue)(y_2))`

Substituting the values from the points in the problem gives:

`M = ((color(red)(-5) + color(blue)(4))/2 , (color(red)(12) + color(blue)(-3))/2)`

`M = ((color(red)(-5) + color(blue)(4))/2 , (color(red)(12) - color(blue)(3))/2)`

`M = (-1/2 , 9/2)`

The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(-3) - color(blue)(12))/(color(red)(4) - color(blue)(-5)) = (color(red)(-3) - color(blue)(12))/(color(red)(4) + color(blue)(5)) = -15/9 = -5/3`

Let's call the slope of a perpendicular line: `m _p`

The formula for the slope of a perpendicular line is:

`m_p = -1/m`

Substituting gives:

`m_p = -1/(-5/3) = 3/5`

Now that we have the slope of the perpendicular line and a point on the line (the midpoint of the line segment) we can use the point-slope formula to write an equation for the line. The point-slope form of a linear equation is: `(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))`

Where `(color(blue)(x_1), color(blue)(y_1))` is a point on the line and `color(red)(m)` is the slope.

Substituting the slope and the values from the mid-point gives:

`(y - color(blue)(9/2)) = color(red)(3/5)(x - color(blue)(-1/2))`

`(y - color(blue)(9/2)) = color(red)(3/5)(x + color(blue)(1/2))`

Answer 2

`3x-5y=-23`

Explanation:

Given the points:
`color(white)("XXX")(-5,12)" and "(4,-3)`

a. The slope between the given points is
`color(white)("XXX")m=(Deltay)/(Deltax)=(12-(-3))/(-5-4)=15/(-9)=-3/5`

b. The slope of any line perpendicular to this is
`color(white)("XXX")hatm=-1/m=5/3`

c. The midpoint between the given points is
`color(white)("XXX")(hatx,haty)=((-5+4)/2,(12+(-3))/2)=(-1/2,9/2)`

d. The equation, in slope-point form, for the perpendicular through the midpoint is
`color(white)("XXX")y-haty=hatm(x-hatx)`

`color(white)("XXX")y-9/2=3/5(x-(-1/2))`

e. Converting to standard form:
after multiplying both sides by `10` to get rid of the fractions:
`color(white)("XXX")10y-45=6x+1`

rearranging into the standard form: `Ax+By=C`
`color(white)("XXX")6x-10y=-46`
and simplifying by dividing all terms by `2`
`color(white)("XXX")3x-5y=-23`