[Guest]

I have a triangle and I have a quadrilateral as well, now I state that if I cut out these I will get two perfectly identical objects.

Is this possible?

NOTE: Assume that I make a flawless cut

Please use spoiler!

Edited April 10, 2010 by rookie1ja
poll removed

[superprismatic]

It is impossible. Edges are edges, after all!

[Guest]

It is impossible. Edges are edges, after all!

Think again! it's possible; I thought the triangle with two right angles will be harder, however it's easy if you know the spherical geometry, by the way that problem is also not solved.

Edited April 11, 2010 by det

[superprismatic]

OK, suppose one draws a triangle on a hollow sphere.

Cut out a quadrilateral having the following boundaries:

two parallel latitudinal cuts -- one which cuts along one

side of the triangle and the other, parallel to the first,

and cutting the vertex of the triangle opposite the side

just cut; and two cuts along the two remaining edges of

the triangle. The pieces we now have are two bowl-shaped

pieces, a quadrilateral band, and the triangle. So, by

cutting out the quadrilateral and letting it fall away

along with the bowl shapes, we have a (spherical) triangle

remaining. Voilà! We get the same result as if we had

just cut out the spherical triangle and let the rest of

the sphere fall away.

[Guest]

OK, suppose one draws a triangle on a hollow sphere.

Cut out a quadrilateral having the following boundaries:

two parallel latitudinal cuts -- one which cuts along one

side of the triangle and the other, parallel to the first,

and cutting the vertex of the triangle opposite the side

just cut; and two cuts along the two remaining edges of

the triangle. The pieces we now have are two bowl-shaped

pieces, a quadrilateral band, and the triangle. So, by

cutting out the quadrilateral and letting it fall away

along with the bowl shapes, we have a (spherical) triangle

remaining. Voilà! We get the same result as if we had

just cut out the spherical triangle and let the rest of

the sphere fall away.

much more simpler than you think, you do not have to manipulate with remaining and falling pieces

You can easily solve the problem on an euclidean plane, btw the solution on spherical space is amost the same

Edited April 11, 2010 by det

[superprismatic]

much more simpler than you think, you do not have to manipulate with remaining and falling pieces

You can easily solve the problem on an euclidean plane, btw the solution on spherical space is amost the same

I don't believe that. If you cut out a quadrilateral and throw away what's left, you've got a

quadrilateral -- not a triangle. So, what you throw away matters.

[Guest]

I don't believe that. If you cut out a quadrilateral and throw away what's left, you've got a

quadrilateral -- not a triangle. So, what you throw away matters.

Are you sure? Do you want to bet?

Imagine a quadrilateral with one 180° inner angle

[superprismatic]

Are you sure? Do you want to bet?

Imagine a quadrilateral with one 180° inner angle

Ha! Aren't you playing a little fast and loose with definitions? I guess that's

OK, but it does make the problem more of a riddle than a math puzzle! Thanks,

I enjoyed thinking about this especially while I was doing my boring exercises.

And I came up with an alternate, albeit complicated, solution to yours!

[Guest]

Ha! Aren't you playing a little fast and loose with definitions? I guess that's

OK, but it does make the problem more of a riddle than a math puzzle! Thanks,

I enjoyed thinking about this especially while I was doing my boring exercises.

And I came up with an alternate, albeit complicated, solution to yours!

I think it does not violate any definition, that's the reason why I didn't use the congruent word.

Congratulation for your solution!

Math problems are too easy if you have to follow the ordinary ways.

Edited April 11, 2010 by det

[Guest]

I call shenanigans. Playing fast and loose with the definitions as you did makes using words like trangle and quadrilateral meaningless. You could have easily said, "I have a shape and I have another shape as well, now I state that if I cut out these I will get two perfectly identical objects.

Is this possible?"

[Guest]

I guess it depends on which quadrilateral definition you accept, if your definition said that any three points can’t be on the same line you are right, but I found a lot of definitions with lower requirements (ex. a four-sided plane closed figure; four coplanar line segments linked end to end to create a closed figure).

[Guest]

... ... ... ... ... ...

[Glycereine]

I guess it depends on which quadrilateral definition you accept, if your definition said that any three points can’t be on the same line you are right, but I found a lot of definitions with lower requirements (ex. a four-sided plane closed figure; four coplanar line segments linked end to end to create a closed figure).

The definition "four-sided" implies that two sides aren't end to end (180 degrees) or it is only a 3-sided figure. I have a hard time with this definition as well

[Guest]

As usual, I am out of my depth here. I think you all know what the answer is and just arguing acceptable definitions....

Is the answer

identical triangles can be put together to make a quadrilateral?