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May 17th, 2012, 04:47 AM   #1
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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.
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May 17th, 2012, 07:57 AM   #2
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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.
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May 18th, 2012, 02:57 AM   #3
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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...
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May 19th, 2012, 03:44 PM   #4
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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.
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May 20th, 2012, 02:48 PM   #5
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Re: Paper problem

a scalene triangle can be cut into 4 isoceles triangles: first cut it into two right-angled triangles.
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May 22nd, 2012, 08:23 PM   #6
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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?
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May 23rd, 2012, 02:29 AM   #7
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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.
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May 23rd, 2012, 02:32 AM   #8
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Re: Paper problem

Quote:
Originally Posted by mehoul
a triangle with to equal sides.
TWO equal sides
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May 23rd, 2012, 08:37 AM   #9
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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?
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May 23rd, 2012, 10:42 AM   #10
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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?
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