Insufficient Altitude

Date: 10/09/2003 at 09:30:33
From: Becky
Subject: the perimeter of a triangle

The base of a triangle is 1200 ft.  The altitude is 500 ft.  What is
the length of the third side?

I am studing for the Praxis I test for my teaching certificate and
this was a sample question.  I am afraid everything I learned in
grammar school has left my brain....

Date: 10/09/2003 at 10:09:37
From: Doctor Edwin
Subject: Re: the perimeter of a triagle

Hi, Becky.

I see two things you should know.  First, altitude is not the name for 
one of the sides of the triangle.  If you draw the triangle with one 
of the lines parallel to the ground (horizontal) and call that the 
base, then the altitude is the distance from that line to the 
opposite vertex.  Confused?  Here, I'll draw one:



           *
          *|*
         * | *
        *  |  *
       *   |   *
      ***********

Okay, the bottom line is the "base" and the vertical line is the
"altitude".  So they haven't given you two sides, they've given 
you one side and one other fact about the triangle.  But it's not 
enough to tell you the perimeter.  Here are three triangles with the 
same altitude and base:

             *        
            *|*        
           * | *      
          *  |  *     
         *   |   *    
        *****|*****  

        *        
        * *      
        *   *    
        *     *  
        *       *
        ***********

   *
   |* *
   | *    *
   |  *      *
   |   *        *
   |    ***********

(I drew the altitude in on the top and bottom triangles, so don't be 
confused by the extra vertical line--it's not part of the triangle.)

The perimeter of the top triangle is smaller than the other two, and 
the perimeter of the bottom triangle is larger than the other two. 
Can you see why?

The problem is that we don't know any of the angles in your triangle, 
so we can't tell how the lines are put together. 

The middle triangle above is a special one, called a right triangle.
That square angle at the bottom left is a right angle, 90 degrees.  If
we know that we're dealing with a right triangle, there are two ways
it helps us.  First of all, we now know the lengths of TWO of the
sides, since one of the sides is the same as the altitude. 

Second, if we know how big that angle is, we can tell how big the
opposite side is.  The famous Pythagorean Theorem is a formula that
gives us the length of the third side of a trinagle if we know the
other two.

So my guess would be that either

1) The question is in error. There is no way to calculate 
   the perimeter of a triangle given just the altitude and base. 

Or

2) The questioners meant to say a right triangle, rather than 
   just a triangle.

If it turns out to be (2), and you can't figure out how to use the 
Pythagorean Theorem to get the length of the third side, write back 
and I'll help you out with that.

Good luck on your teaching certificate!

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