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/