Subsets of Shapes

Date: 01/27/2004 at 14:28:59
From: Christine
Subject: Squares and rectangles

What is the relationship between a square and a rectangle?

Date: 01/28/2004 at 12:01:41
From: Doctor Ian
Subject: Re: Squares and rectangles

Hi Christine,

It's sort of like the relationship between an equilateral triangle and
an isosceles triangle.  Let's look at the definitions:

  isosceles triangle:    at least two sides are the same length

  equilateral triangle:  all three sides are the same length

Now, if we have an equilateral triangle, say with sides of length 5,
5, and 5, is there any way for that to NOT be an isosceles triangle? 
No, because if all three sides are the same, then at least two of them
will be the same.  So every equilateral triangle will also be an
isosceles triangle.

Is the reverse true?  No, because we can easily make an isosceles
triangle with sides of length 5, 5, and 6; or 5, 5, and 9; and so on.  

So if we imagine all the triangles that there could ever be, some of
them will be isosceles, 

   +---------------------------------+
   | triangles                       |
   |                                 |
   |  +----------------------+       |
   |  | isosceles triangles  |       |
   |  |                      |       |
   |  +----------------------+       |
   |                                 |
   +---------------------------------+

and some will be equilateral; but every equilateral triangle will also
be isosceles, just as every isosceles triangle is also a triangle:

   +---------------------------------+
   | triangles                       |
   |                                 |
   |  +----------------------+       |
   |  | isosceles triangles  |       |
   |  |                      |       |
   |  |  +---------------+   |       |
   |  |  |  equilateral  |   |       |
   |  |  |  triangles    |   |       |
   |  |  +---------------+   |       |
   |  |                      |       |
   |  +----------------------+       |
   |                                 |
   +---------------------------------+

Can you draw a similar diagram that shows the relationship of
quadrilaterals to rectangles to squares?  How would you add
parallelograms to that diagram? 

If you're looking for a succinct way to characterize this
relationship, take a look at 

  What are Sets and Subsets?
    http://mathforum.org/library/drmath/view/52398.html 

Does this help? 

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