Lesson Forming a rectangle from a square by using dissection

Source code of 'Forming a rectangle from a square by using dissection'

This Lesson (Forming a rectangle from a square by using dissection) was created by by math_helper(2461)

: View Source, Show
About math_helper: MSEE, who has a strong interest in helping others understand concepts related to mathematics.

Here is a problem that was asked recently.  Given a square of material, how can one cut that square and rearrange the pieces to make a rectangle?   The rectangle shall have length equal to three times the width, and ideally uses all of the material from the square.

I was initially lost on this problem.  I found a resource on the web that showed how to make a square from a rectangle and adapted the reverse of that procedure.  However, I had failed to realize it was only for rectangles where length is less than twice the width,  and we wanted a rectangle where length was three times the width.

I continued to research this problem as it seemed so foreign to me:  cutting something and re-arranging it does not seem to fit the normal "transformation" where one draws a shape and morphs it into another on paper.   It turns out, for rectangles up to         l < 4w,  you can use the following procedure.  I adapted the solution from http://mathafou.free.fr/pbg_en/sol110.html  where they go from a rectangle to a square.

{{{
 drawing(400,400, -5,20, -5,20,
      rectangle(0,0,10,10),
      line( 0,10, 10, 4.23 ),
      line( 0,5.77,  7.32,5.77),
      locate(2,7,A),
      locate(7,8,B),
      locate(3,3,C),
      locate(10.2,7, w),
      locate(10.2, 3, h-w),
      locate(0.2, 3, w),
      locate(0.2, 8, h-w),
      locate(10.2,10.2, "(h,h)"),
      locate(-1,-0.5, "(0,0)")
 )
}}}
�

� Cut along the lines and rearrange as shown below �
�

{{{
 drawing(400,400, -5,20, -5, 20,
      rectangle(0,0,  17.32, 5.77),
      line( 5.77,5.77, 17.32, 0),
      line( 10,0,  10, 3.7),
      locate(12, 2, A),
      locate(14,  4, B),
      locate(3,3,C)
 )

}}}