Associated Topics || Dr. Math Home || Search Dr. Math

### Triangle Inequality Theorem

```
Date: 03/09/2001 at 16:30:56
From: Cprime Realista
Subject: Non-isosceles triangle

The lengths of the sides of a non-isosceles triangle, in size order,
are 5, x, and 15. What are all possible integral values of x?

My guess would be that the number is somewhere between 6 and 14.
```

```
Date: 03/13/2001 at 09:58:13
From: Doctor Patti
Subject: Re: Non-isosceles triangle

Hello Cprime,

Thank you for writing to Dr. Math.

There is a theorem in geometry (the triangle inequality theorem) that
states: The sum of the lengths of any two sides of a triangle must be
greater than the length of the third side.

Let's take a look at an "open" triangle with sides x, y, and z

/
x /                 \
/                   \  y
/                     \
------------------------
z

To see the reasoning behind the theorem, think of sides x and y
'folding' down toward side z. If the sum of the lengths of x and y is
not greater than the length of z, they will lie flat on z, and this
would not make a triangle.

So, from the theorem

x + y > z
y + z > x
x + z > y

Let's look at a triangle with sides 2, 8, and x. What are all possible
integral values of x?

x would need to be greater than 6; otherwise sides with 2 and 6
would lie flat on the side with length 8.

x would also need to be less than 10; otherwise sides 2 and 8 would
lie flat on the side with length 10.

Therefore,

6 < x < 10

or

x is between, but does not include, 6 and 10

When working with this type of problem I think to myself: the length
of x, the third side, needs to be LESS than than the sum of the two
sides and GREATER than the difference of the two sides.

I hope this helps.  Please write back if you have any other questions.

- Doctor Patti, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search