Proving Diagonals Perpendicular

Date: 07/28/2003 at 09:10:05
From: Dan16etta
Subject: Perpendicular diagonals

Given the points a(-4,1), b(2,3), c (4,9) and d (-2,7), show that 
quadrilateral abcd is a parallelogram with perpendicular diagonals.

Date: 07/28/2003 at 13:35:48
From: Doctor Barrus
Subject: Re: perpendicular diagonals

First I'm going to draw a rough picture of the four points, so I can 
see what we're talking about.

            |       * c
            |
      d *   |
            |
            |
            |
            |   * b
            |   
  a *       |   
------------+-----------

We want to show that the two diagonals of the quadrilateral are 
perpendicular. In other words, if we drew a line passing through a 
and c, and another line passing through b and d, those two lines 
would be perpendicular to each other.

Now how can you tell if two lines are perpendicular? One way, most 
likely the way you'll want to use here, is to look at their slopes. 
If the product of two lines' slopes is equal to -1, then the two 
lines are perpendicular.

So let's find the slopes of the two diagonals.

            rise
Slope ac = ------
            run

             9 - 1
         = ----------
            4 - (-4)

         = 8/8

         = 1

Similarly (you'll want to work this out for yourself), the slope of 
line bd is -1. Now if we take the two slopes and multiply them, we get

slope AC * slope BD = 1 * -1 = -1

and since the product is -1, we know that these two lines are 
perpendicular. Make sense?

I hope that helps. If you have further questions, feel free to write 
back. Good luck!

- Doctor Barrus, The Math Forum
  http://mathforum.org/dr.math/