Two geometry problems -- lines and straightedges

[LittleRookie]

Homework Statement

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Relevant Equations

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Hello all, I need some help on two exercises from Kiselev's geometry, about straight lines.

Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down.

I would draw the first line, then flip the straightedge and draw the second line over the first. The two lines should coincide nicely iff the straightedge is straight. Because, this shows that there is no "unevenness" or "bumps" on the edge of the straightedge. There would be gaps between the two lines if there are "unevenness/bumps" on the edge of the straightedge.

I know my explanation is pretty flawed, but this is the best I could think of.

Ex 8: Fold a sheet of paper and, using ex 7, check that the edge is straight. Can you explain why the edge of a folded paper is straight?

Ex 8 is marked as more difficult by the author. I'm completely clueless about this exercise.

Please provide insights and help me with these two exercises. I'd appreciate if they are more of an "experimental approach" than theoretical because exercises 7 and 8 are arranged in between the introduction and first chapter of the book.

Thank you. :)

 

[symbolipoint]

That's a very good, clever description. Hard to say in any formal way how any of that might be improved. Your natural intuition should tell you that folding a paper will form a very straigtht-edge as the fold; you would then be able to use the external folded edge as a, somewhat dentable, straight-edge.

 

[LittleRookie]

That's a very good, clever description. Hard to say in any formal way how any of that might be improved. Your natural intuition should tell you that folding a paper will form a very straigtht-edge as the fold; you would then be able to use the external folded edge as a, somewhat dentable, straight-edge.

Can you provide an explanation for exercise 8? I prefer logical explanation than a "hunch" or intuition.

 

[BvU]

It's your exercise, we don't want to rob you from it ...

PF gives hints or guiding questions at best, see guidelines.

Try to turn it around: why can't a fold be curved ? What would be needed to make it curved ?

 

[LittleRookie]

I'm still completely clueless about the fold.

 

[SammyS]

I'm still completely clueless about the fold.

The hint from Ex 7 may help by using that hint to determine if a line you draw with the folded paper is straight.

 

[LittleRookie]

The hint from Ex 7 may help by using that hint to determine if a line you draw with the folded paper is straight.

I have an idea about exercise 8.

Making a fold on a piece of paper, is the same as cutting the piece of paper along the fold into two, and then flipping one onto the top of the another.

If I were to cut a piece of paper along a curve and then flipping one onto the other, the cut sides will not fit exactly.

Wherelse, cutting along a straight line and then flipping one onto the other, the cut sides fit exactly.

Conversely, thinking folding a piece of paper as momentarily separating the piece of paper into two and placing one on top of the other, the "separated" sides fit exactly, which means the fold must be straight.

 

[BvU]

I like it ! You might want to add 'without shifting' -- or else a zig-zag line might qualify too.

Other approaches might be along the lines () of

two points -- one line only

three points on one line means you can shift one point along the line and make it coincide with one of the other two points

substitute fold for line and the same statements hold.

( desclaimer: I don't really know when a claim is really a proof -- or when it's just an axiom in another form -- interesting discussions possible!)

 

[SammyS]

I have an idea about exercise 8.

Making a fold on a piece of paper, is the same as cutting the piece of paper along the fold into two, and then flipping one onto the top of the another.

If I were to cut a piece of paper along a curve and then flipping one onto the other, the cut sides will not fit exactly.

Where as, cutting along a straight line and then flipping one onto the other, the cut sides fit exactly.

Conversely, thinking folding a piece of paper as momentarily separating the piece of paper into two and placing one on top of the other, the "separated" sides fit exactly, which means the fold must be straight.

What if you just leave the sheet of paper folded over, without cutting it?

There is a way to "flip" this straight-edge^* over, in such a way that edge doesn't move. It seems to me that coming up with this way of flipping the straight-edge solves the problem.

(^*) The folded sheet of paper.

 

[LittleRookie]

What if you just leave the sheet of paper folded over, without cutting it?

There is a way to "flip" this straight-edge^* over, in such a way that edge doesn't move. It seems to me that coming up with this way of flipping the straight-edge solves the problem.

(^*) The folded sheet of paper.

Is it this way of flipping as described in the picture?

 

[SammyS]

Is it this way of flipping as described in the picture?View attachment 244507

Yes.

So, does any part of the blue line move when doing this?

 

[LittleRookie]

It didn't move a single bit. I understand now, thanks.

 

[MidgetDwarf]

Do you know about Saccheri Quadrilaterals and the associated Saccheri Quadrilateral of a Triangle? If so, there is a problem similar, although a bit different, to the one you are attempting.

 

[kuruman]

I don't know much about Saccheri quadrilaterals, but I do know that after establishing in exercise 7 that my ruler is straight, I can use it to draw a guaranteed straight line between two different points. Then I would repeat the procedure in exercise 7 using the folded sheet as the ruler to verify that it is straight.