Sides of Similar Triangles

Date: 06/11/98 at 22:13:25
From: michael gonzalez
Subject: Geometry

The sides of a triangle are 24, 16, and 12. The shortest side of a 
similar triangle is 6. Find the longest side of this triangle.

Date: 06/12/98 at 23:04:19
From: Doctor Sorelle
Subject: Re: Geometry

Dear Michael,

Why don't we draw these two triangles to get you started:

         |\                   |\
         | \                  | \ x
         |  \               y |  \
      16 |   \  24            |___\
         |    \                 6
         |     \
         |______\
            12

So 6 is the shortest side of a triangle that is similar to the one 
whose shortest side is 12. We can see, then, that these two sides must 
be related. 

Similar triangles have the same angles and are exactly the same except 
that the sides are shorter, but the sides have to be proportionally 
shorter or longer. This means that if one side of a similar triangle 
is 5 times as long, all of the other sides will be 5 times as long as 
well.

So what if we have a small triangle whose sides are 3, 4, and 5 and a 
bigger similar triangle whose longest side is 15?

         |\                   |\
         | \                  | \ 5
         |  \               4 |  \
      x  |   \  15            |___\
         |    \                 3
         |     \
         |______\          Triangle 2
            y

        Triangle 1         


Do you know how we would find the side that corresponds with 3 (the 
shortest side, y)?

Well, we know that 15 is 3 times as big as 5. If these are similar 
triangles and all of the sides are proportionally larger, then y would 
have to be 3 times as big as 3, right? That would make y equal to 9.

Another way of looking at this is through ratios. We could set up a 
ratio:

   1st small side   1st big side       
   -------------- = ------------     
   2nd small side   2nd big side

This would be:

   y   15       
   - = ---     
   3    5

These would be equal because the sides are proportionally equal. We 
can see that both of these fractions are equal to 3, the three times 
bigger that the 1st triangle is than the 2nd. If you wanted to find y 
you could use algebra to solve the equation.

Now do you think that you can go back to your problem and find out 
what the longer side is for the similar triangle?

Good luck!

-Doctor Sorelle,  The Math Forum
Check out our web site! http://mathforum.org/dr.math