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### 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/
```
Associated Topics:
High School Basic Algebra
High School Coordinate Plane Geometry
High School Geometry
High School Triangles and Other Polygons
Middle School Algebra
Middle School Geometry
Middle School Triangles and Other Polygons

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