How do you find the slope of a line perpendicular to a slope of a line is 1/3?

1 Answer
George C.
May 20, 2015

In general, if the slope of a line is #m#, then the slope of any perpendicular line will be #-1/m#. So in your case the slope of any perpendicular line would be #-1/(1/3) = -3#.

To see that, consider a line given by

#y = mx+c#

If you reflect that line in the line #y=x#, you get a line whose equation is the same, but with #x# and #y# swapped:

#x = my+c#

If you then reflect that line in the #x# axis, you are basically reversing the sign of the #y# coordinate, so you get a line with equation:

#x = -my+c#

The total result of these two geometric operations is to rotate the original line through a right angle (try it yourself with a square of paper).

Now we can rearrange this new line's equation into slope intercept form as follows:

Add #my# to both sides:

#my + x = c#

Subtract #x# from both sides:

#my = -x + c#

Divide both sides by #m#:

#y = (-1/m)x + c/m#

Notice the new slope is #-1/m#.

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