Length of a Triangle's Sides

Date: 23 Jan 1995 14:05:32 -0500
From: Scott Rappard
Subject: Braemore School Antigonish N.S. Canada

Dear Dr. Math,

My name is Matthew Rappard and I have a problem for you to solve.

The lengths of the three sides of a triangle could be
A 0,1,2    B 1,2,3    C 2,3,4   D 2,4,6

I got this problem from my teacher.  My class could not get it, and the 
teacher can't get it either.  We know the answer is C, but why?

Thanks for your help
Matthew Rappard

Date: 24 Jan 1995 01:11:21 GMT
From: Dr. Math
Subject: Re: Braemore School Antigonish N.S. Canada

Hello there!

Well, C is indeed the right answer.  Think about the following property of
triangles:
                            A
                           /\
                          /  \
                         /    \
                        /      \
                       /        \
                      /          \
                     /            \
                    /              \
                   /                \
                  /                  \
                 /                    \
                /                      \
               /                        \
              /                          \
             /                            \
            /                              \
           /                                \
          /                                  \
          ____________________________________
          B                                  C

Look at the distance between B and C.  How far is it?  You can measure if
you want.  That's the minimum possible distance along any path from B to
C.  Now what would happen if you had walked along a different path from B
to C, like starting at B, walking to A, and then walking to C?  It would
be a longer trip, right?  So what we've said is that the distance from B
to C is less than the distance from B to A plus the distance from A to C. 
Or in equation notation,

BC < BA + AC.  

You'll also notice that there was nothing special about BC, the same thing
happens with AB or AC, and we have the following relations:

BC < BA + AC  (this is from before)
BA < BC + AC
AC < BC + BA

These are the three relations that govern the sides of a triangle:  if
you add the lengths of any two sides, you'll get something bigger than the
length of the third side.  

Now look at your possibilities for sides, and see whether you can figure
out why they chose them like they did.  Write back if you have more
questions!

-Ken "Dr." Math