## Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

# The kindest (shortest) cut of all

Best Answer superprismatic, 27 February 2014 - 08:56 PM

Spoiler for that nice hint makes it look like

Go to the full post

6 replies to this topic

### #1 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 25 February 2014 - 10:39 PM

Here is a piece of plywood in the shape of an isosceles triangle.

The side lengths are 1, 1, sqrt(2) units.

Quick and dirty representation:

A

|  \

|     \

|        \

|           \

B--------------C

The angle at B is a right angle.

We'd like to cut this into two pieces of equal area.

There are many ways to do this with a single cut.

Which cut has the shortest distance?

• 0

Vidi vici veni.

### #2 TimeSpaceLightForce

TimeSpaceLightForce

Senior Member

• Members
• 705 posts
• Gender:Male

Posted 26 February 2014 - 10:50 PM

Spoiler for cutting lines

• 0

### #3 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 27 February 2014 - 10:11 AM

Spoiler for cutting lines

With a careful reading of the OP can you do slightly better?

• 0

Vidi vici veni.

### #4 superprismatic

superprismatic

Not just Prismatic

• Moderator
• 1281 posts
• Gender:Male

Posted 27 February 2014 - 08:56 PM   Best Answer

Spoiler for that nice hint makes it look like

• 0

### #5 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 27 February 2014 - 09:01 PM

Kudos to both TSLF and SP.

• 0

Vidi vici veni.

### #6 superprismatic

superprismatic

Not just Prismatic

• Moderator
• 1281 posts
• Gender:Male

Posted 27 February 2014 - 10:57 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest?  Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter.  The possibilities are endless.

• 0

### #7 bonanova

bonanova

bonanova

• Moderator
• 6153 posts
• Gender:Male
• Location:New York

Posted 27 February 2014 - 11:41 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest?  Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter.  The possibilities are endless.

Yes, it's called the isoperimetric problem, generally stated,

what is the curve of constant length that encloses the greatest area?

Equivalently, what shape of constant area has the smallest perimeter?

The answer is a circle. A 45o arc can be reflected 7 times into a full circle.

Symmetry demands this answer, although rigorous proofs abound.

Here is a general discussion.

In three dimensions (think soap bubble) a sphere encloses more volume

than any other closed surface of the same area.

isoperimetric = iso (same) perimetric (perimeter).

• 0

Vidi vici veni.

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users