My Math Forum Paper problem

 Number Theory Number Theory Math Forum

 May 17th, 2012, 04:47 AM #1 Member   Joined: Dec 2009 Posts: 65 Thanks: 0 Paper problem Hello! I have an interesting problem, and I don't have any idea how to solve it... It is possible to cut a triangle-shaped paper to n (2, 3, 4, 5, ...) isosceles triangle-shaped piece of papers? Many thanks, Crouch.
 May 17th, 2012, 07:57 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Paper problem The case n = 2 is not possible in general. Since the original triangle must be cut into two triangles (rather than a triangle and a quadrilateral), you must cut from a point to a side (not a side to a side or a point to a point). Suppose the original triangle has sides of length 10, 10, and 1. Then one of the smaller triangles must have a side of length 10, and hence it must have a second of length 10 to be isosceles (since otherwise it must have two equal sides of length > 5, and you don't have that much 'material'.) But this is not possible without using the whole triangle, leaving nothing for the second triangle.
 May 18th, 2012, 02:57 AM #3 Member   Joined: Dec 2009 Posts: 65 Thanks: 0 Re: Paper problem Thanks, understood. But can we say something about the very general case? The case of n? I think it's very difficult...
 May 19th, 2012, 03:44 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond Re: Paper problem I don't think any n is possible if the sheet of paper is initially a scalene triangle. If it is initially an isosceles triangle then there any n > 2 isosceles triangles that can be formed, with size as the only limiting factor.
 May 20th, 2012, 02:48 PM #5 Senior Member   Joined: Feb 2012 Posts: 144 Thanks: 16 Re: Paper problem a scalene triangle can be cut into 4 isoceles triangles: first cut it into two right-angled triangles.
May 22nd, 2012, 08:23 PM   #6
Global Moderator

Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

Math Focus: Elementary mathematics and beyond
Re: Paper problem

Quote:
 Originally Posted by mehoul a scalene triangle can be cut into 4 isoceles triangles: first cut it into two right-angled triangles.
A 30-60-90 triangle is scalene and can be divided into two isosceles triangles, yes, but can any scalene right triangle be divided into two isosceles triangles? I don't think so - only a 30-60-90 triangle has that property.

Can you provide another counter-example?

 May 23rd, 2012, 02:29 AM #7 Senior Member   Joined: Feb 2012 Posts: 144 Thanks: 16 Re: Paper problem mmmh, I may not understand correctly the problem... to me an isosceles triangle is a triangle with to equal sides. And to divide a triangle in two triangles is to draw a line from one vertex to the opposite edge. If I am right, then join the vertex where the right angle to the middle point of the opposite edge. This middle point is equidistant from the three vertices, hence the two triangles are isosceles.
May 23rd, 2012, 02:32 AM   #8
Senior Member

Joined: Feb 2012

Posts: 144
Thanks: 16

Re: Paper problem

Quote:
 Originally Posted by mehoul a triangle with to equal sides.
TWO equal sides

May 23rd, 2012, 08:37 AM   #9
Global Moderator

Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

Math Focus: Elementary mathematics and beyond
Re: Paper problem

Quote:
 Originally Posted by mehoul mmmh, I may not understand correctly the problem... to me an isosceles triangle is a triangle with to equal sides. And to divide a triangle in two triangles is to draw a line from one vertex to the opposite edge. If I am right, then join the vertex where the right angle to the middle point of the opposite edge. This middle point is equidistant from the three vertices, hence the two triangles are isosceles.
Yes, isosceles is two equal sides.

The construction you give above must be a 30-60-90 triangle, no?

 May 23rd, 2012, 10:42 AM #10 Member   Joined: Dec 2009 Posts: 65 Thanks: 0 Re: Paper problem Mehoul, you are not right. In general, you can't cut a triangle to two isosceles triangles. In the second comment is proved, that we can't cut a triangle to two isosceles triangles. Any idea for the general case?

 Tags paper, problem

crease length calculus problem

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post jaredbeach Calculus 2 December 14th, 2011 05:25 AM Deb_D Advanced Statistics 3 November 23rd, 2010 10:07 AM SH-Rock Calculus 0 October 11th, 2010 10:23 AM Cola Algebra 4 October 21st, 2008 05:31 AM bigli Real Analysis 3 May 22nd, 2007 11:00 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top