# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Tue, 21 May 2013 23:59:18 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 6 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Segments and Areas in a Unit SquarePosted: Mon, 11 Jun 2012 00:26:47 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Segments and Areas in a Unit SquarePosted: Mon, 11 Jun 2012 03:51:37 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?

Use a baby version of the isoperimetric inequality
Spoiler:
With a fixed perimeter and adjacent sides perpendicular, the figure with the biggest area is the square (Prove it!)

_________________

Top

 Post subject: Re: Segments and Areas in a Unit SquarePosted: Mon, 11 Jun 2012 16:48:11 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7949
Location: Lexington, MA
Hello, rdj5933mile5math64!

I have an intuitive approach . . .

Quote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments.
Prove that one of these regions has an area that is greater than or equal to units.

Consider using 8 vertical lines and 10 horizontal lines, equally spaced.
The unit square is divided into (at most) 99 squares.
Hence, each square has area of at least

Consider using 9 vertical lines and 9 horizontal lines, equally spaced.
Then the unit square is divided into (at most) 100 squares.
Hence, each square has area of at least

If the lines are not equally spaced,
. . some rectangles have an area less than
. . and others have areas greater than

Top

 Post subject: Re: Segments and Areas in a Unit SquarePosted: Mon, 11 Jun 2012 17:05:42 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Soroban wrote:
If the lines are not equally spaced,
some rectangles have an area less than
and others have areas greater than

Err... nobody said the segments each have to span the entire width/height of the square, and moreover, you can have shapes other than rectangles.

_________________

Top

 Post subject: Re: Segments and Areas in a Unit SquarePosted: Mon, 11 Jun 2012 17:07:32 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
outermeasure wrote:
Soroban wrote:
If the lines are not equally spaced,
some rectangles have an area less than
and others have areas greater than

Err... nobody said the segments each have to span the entire width/height of the square, and moreover, you can have shapes other than rectangles.

I think that was the point behind "intuitive".

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Segments and Areas in a Unit SquarePosted: Tue, 12 Jun 2012 18:44:03 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
rdj5933mile5math64 wrote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?

Use a baby version of the isoperimetric inequality
Spoiler:
With a fixed perimeter and adjacent sides perpendicular, the figure with the biggest area is the square (Prove it!)

Got it! Thank you!

_________________
math puns are the first sine of madness
-JDR

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 6 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous