SOLUTION: Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.

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Question 1207909: Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.

Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.
~~~~~~~~~~~~~~~~~~~ 

The slope of a straight line is a value of the tangent function of the angle between this line and x-axis.

So, we can write  m%5B1%5D = tan%28alpha%29,  m%5B2%5D = tan%28beta%29,  where  alpha  and  beta  are the angles 
that these straight line form with x-axis.


The angle between these straight lines is  beta-alpha.  Let's calculate the tangent of this angle


    tan%28beta-alpha%29 = %28tan%28beta%29-tan%28alpha%29%29%2F%281%2Btan%28beta%29%2Atan%28alpha%29%29.


The denominator  1%2Btan%28beta%29%2Atan%28alpha%29  is  1+%2B+m%5B1%5D%2Am%5B2%5D = 1 + (-1) = 0,  so  tan%28beta-alpha%29  is not defined.


But the tangent function of an angle is not defined if and only if the angle is +/- 90 degrees.


Thus we proved that the angle between our lines is +/- 90 degrees.


It means that the lines are perpendicular.

At this point, the proof is complete and the problem is solved.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Get a sheet of (graph) paper.
You can do this digitally with something like GeoGebra or similar.
Something like MS Paint works as well.

Plot a point somewhere along the left edge and somewhere in the middle.
The exact location doesn't matter.
Let's label this as point M.

From M, move up 'a' units to arrive at another point I'll label N.
From N, move b units to the right to land on point P.
This forms right triangle MNP.
line MP has slope a/b

Return to point M.
Move down b units to get to point Q.
From Q, move 'a' units to the right to land on point R.
This forms right triangle RQM.
line MR has slope -b/a
Refer to the diagram below.


Key takeaways:
line MP has slope a/b
line MR has slope -b/a

The product of these slopes is -1
This is because the 'a's cancel, and so do the 'b's.
All that's left is the minus sign which yields the -1.

--------------------------------------------------------------------------

Claim: lines MP and MR are perpendicular.

Proof:

Triangle MNP and triangle RQM represent right triangles.
The 90 degree angles are located at points N and Q.

Recall that the acute angles of a right triangle are complementary.
They add to 90 degrees.
For triangle MNP, we have: (angle NMP) + (angle NPM) = 90
This is the sum of the blue and red angles of triangle MNP.
See the diagram below.


Not only do we have right triangles, it turns out they are congruent triangles.
How can we prove congruence? By use of the SAS (side angle side) congruence theorem.
One triangle is a rotated clone of the other.

Since these triangles are twin clones, they must of course share the same congruent corresponding parts.
It's like saying two houses are identical, so their front doors must be identical.
"house" in this analogy would be the entire triangle.
"front door" would represent a piece of the triangle, e.g. one of the acute angles.

The punchline I'm trying to get to here is that
angle NPM = angle QMR
These angles are marked in red in the diagram below.


So we know these two things
(angle NMP) + (angle NPM) = 90
angle NPM = angle QMR
I'll refer to these as equations (1) and (2) in the order shown above.


Let's wrap up the proof.
(angle NMP) + (angle PMR) + (angle QMR) = 180 degrees
(angle NMP) + (angle PMR) + (angle NPM) = 180 degrees ..... use equation (2)
(angle NMP + angle NPM) + (angle PMR) = 180 degrees
(90) + (angle PMR) = 180 degrees ....................................... use equation (1)
angle PMR = 180 - 90
angle PMR = 90

We have shown that lines MP and MR are perpendicular to each other, because they meet up at a 90 degree angle.
Recall that we have MP and MR with slopes a/b and -b/a respectively.
These slopes multiply to -1.
Therefore, any pair of perpendicular nonvertical lines have their slopes multiply to -1.

The "nonvertical" requirement is to avoid undefined slopes.


Notice that a,b are both nonzero.
Otherwise we run into a division by zero error.

Diagram

The diagram was made with GeoGebra, but you can use any other alternative you prefer.