All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Rectangle to Square (Posted on 2013-08-13) Difficulty: 2 of 5

A rectangle ABCD is to be cut into three pieces
(with two straight line cuts) and the pieces re-
arranged into a square.

Define the endpoints of the cuts.

Note 1: |AB|/4 ≤ |BC| < |AB|.

Note 2: A straight line cut (as the name implies)
            is a cut along a straight line that
            divides a piece into two pieces.

See The Solution Submitted by Bractals    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Full solution | Comment 7 of 9 |
(In reply to Full solution by Jer)

here is how I analized your solution. First some labeling.  In the first drawing, let the length of the horizontal (top/bottom) side by m and let the length of the vertical (left/right) side be n.  Also, I assumed that m<n, if m=n then any cut works as it is already a square.  Now for the diagonal cut, let end of the cut be x from the top.  Also, let the horizontal cut be y from the top.  Finally, let the length of the horizontal cut be z.  Now from similar triangles we have x/m=y/z thus z=my/x.

Now in the bottom drawing we have the left side as x and the right side as n-y.  Also both the top and bottom are given by m+z=m+my/x.  So for this to be a square we need both
substituting the first in the second we have
n-y=m+my/(n-y)  since y<n (otherwise there can't be a cut) we can multiply through by n-y to get
nowhever since m<n then sqrt(mn)<n and since y<n we have
which gives x=sqrt(mn)<n thus the diagonal cut can happen as well.

So from this analysis I conclude that your solution works for all rectangles.

  Posted by Daniel on 2013-08-16 10:34:57
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information