Question

What is the slope of any line perpendicular to the line passing through `(15,-12)` and `(24,27)`?

Answer 1

`-3/13`

Explanation:

Let the slope of the line passing through the given points be `m`.

`m=(27-(-12))/(24-15)=(27+12)/9=39/9=13/3`

Let the slope of the line perpendicular to the line passing through the given points be `m'`.

Then `m*m'=-1 implies m'=-1/m=-1/(13/3)`

`implies m'=-3/13`

Hence, the slope of the required line is `-3/13`.

Answer 2

The slope of any line perpendicular to the given one is: `-3/13`

Explanation:

The trick is to just remember that if the gradient of the first line is `m` the gradient of the one that is perpendicular to it (normal) has the gradient of `(-1)xx1/m`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Gradient (slope) of the first line")`

Let `m_1` be the gradient of the first line

Then
`m_1=(y_2-y_1)/(x_2-x_1)`

Given that
`(x_1,y_1)-> (15,-12)`
`(x_2,y_2)-> (24,27)`

We have:
`color(blue)(m_1=(27-(-12))/(24-15) color(white)(....)-> color(white)(....) 39/9)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Gradient (slope) of the second line")`

Let `m_2` be the gradient of the second line

Then
`m_2=(-1)xx1/m_1color(white)(....)-> color(white)(....)(-1)xx 9/39`

`color(blue)(m_2= -(9-:3)/(39-:3) =-3/13)`