Find the Length of the Third Side

Date: 08/26/2003 at 10:26:57
From: Susan
Subject: Determining the dimensions of a triangle

If one side of a triangle is 6 and the other is 8, how do you 
determine the length of the third side? 

This was from an IQ test: two cars start at the same point, going in 
opposite directions for six miles; both make a left turn and go 8 more 
miles. How many miles apart are the cars?

Date: 08/26/2003 at 11:16:37
From: Doctor Edwin
Subject: Re: Determining the dimensions of a triangle

Hi, Susan.

Interesting question. In general, there is no way to know the third 
side of a triangle if you know only two. Picture a triangle with two 
sides of fixed length, and picture the third being made of a rubber 
band. You with me? 

Now imagine that the point where the two fixed sides meet is hinged 
so that you can change the angle. If you open it up, the rubber band 
stretches. If you close it, the rubber band shrinks. So the third 
side of a triangle can have a range of sizes.

However, in this problem they expect you to make an assumption. They 
expect you to assume that when the cars turn left they turn at right 
angles to the way they were going.

So their path now looks like this:

  1--------------
                |
                |
                |
                |
                |
                | <- starting point.
                |
                |
                |
                |
                |
                --------------2

If you draw a line between car 1 and car 2, you get two triangles 
that meet at the starting point. The two triangles are mirror images 
of each other (we can prove that, but I think if you sort of stare at 
it for a minute you'll agree). Let's deal with just one triangle, and 
then we can double the length of the side to figure how far apart the 
two cars are:



                |*<- starting point.
                |  *
                |    *     c
              a |       *
                |         *
                |           *
                --------------2
                       b

Now if the angle between a and b weren't a right angle, we'd have to 
use trigonometry to figure out the length of side c. Since it is a 
right angle, we can use the Pythagorean theorem to get the length.

You may remember the Pythagorean theorem as something you had to 
memorize. It says that if you square the length of side a, and add it 
to the square of side b, that will equal the square of side c:

  a^2  +  b^2  =  c^2 (I'm using ^2 to mean "squared" since I can't 
write the little 2's up above the letters).

  6^2  +  8^2  =  c^2

Can you solve it from here? If you're still stuck, write back with the
problem you're having and I'll try to help.

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