A Right Triangle of Points

Date: 01/14/99 at 22:44:50
From: Matt Strenz
Subject: Right triangle

Determine the values of x that would make the points (x,0), (-2,1), and 
(3,4) the vertices of a right triangle.

I have tried the problem but do not understand it. Could you give me 
some help please? Thank you.

Date: 01/15/99 at 13:02:12
From: Doctor Peterson
Subject: Re: Right triangle

Hi, Matt. I think I can point you in the right direction.

The first thing I would do is to graph the points so we can see what 
we have:

              |       C(3,4)
             4+        *
              |
             3+
              |
             2+
   B(-2,1)    |
        *    1+
              |              A(x,0)
      --+--+--+--+--+--+--   *
       -2     0  1     3

We have two points (B and C) nailed down, and one (A) that can slide 
anywhere along the x axis. We have to figure out what values of x will 
give us a right triangle.

The trouble is, there are three ways we could get a right triangle, 
depending on which point is the right angle. So you really have three 
problems to solve.

The easiest cases will be where one of the fixed points B or C is the 
right angle. You can find the slope of the segment BC, and from that 
you know what the slope of AB (or AC) has to be. (Remember that the 
slope of the perpendicular line is the negative reciprocal.) This will 
give you an equation like this to solve:

     1 - 0
    ------ = m
    -2 - x

where m is the slope you calculated.

If A is the right angle, you can make an equation based on the fact 
that AB and AC have to be perpendicular.

That should give you a good start on the problem.

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