Slopes of Perpendicular Lines …

A proof that

 

If you have wondered WHY the slopes of perpendicular lines have to be negative reciprocals, here is an answer.  This proof uses the geometry of the coordinate system (and lots of subscripts like the 1 and 2 above); the property can also be proven with trigonometry.

 


First, let’s start with a general picture of perpendicular lines.

The equation of the lines are  and

 

At the point where the lines intersect, the y values are equal.

Since y1 = y2, we know that

[This is very important in the proof!!]

 

Since we are working with slopes, we need a second point on each line.

Let’s find the y values when the x-coordinate is (x+1).   (We could use any convenient value, like (x+2) or (x-1) …it’s just a bit simpler if we use (x+1) in our work.)

 

(x+1, m1x+m1+b1)
 
            y1: 

 


(x+1, m2x+m2+b2)
 
y2: 

 

Now we have two points on each line, three points all together.  What shape do those points make??   A right triangle!!

hypotenuse
 
Here is a close-up of the right triangle

 

We need to write expressions for the lengths of the two legs and

the length of  the hypotenuse.   This will be based on the distance

formula between two points.

           

The top leg (from the intersection to the next point on y1 ): 

For the bottom leg (from the intersection to the next point on y2  ):

For the hypotenuse:                             *                    

In the third step, we use the fact that the lines intersect – which means that m1x+b1 = m2x+b2.   Since these quantities are equal, subtracting them (m2x+b2 – m1x - b1 ) gives zero – they cancel out.

 

In the right triangle, we apply the Pythagorean property – the sum of the squares of each leg equals the square of the hypotenuse.

                        Leg˛         +   Leg˛    =  Hypotenuse˛

Subtract and from both sides.

Divide both sides by -2
 

(a-b)˛=(a-b)(a-b)

=a˛-2ab-b˛
 
                          

 

There we go!!   If the product of the slopes is –1, then the slopes are negative reciprocals.

 

In other words, when two lines are perpendicular, they form a right triangle.  In order for the right triangle to follow the Pythagorean property, the slopes have to be negative reciprocals.