Proving that all straight lines that bisect a square go through the center point

[quick_maths]

I mean, it‘s basic knowledge and just by drawing this I know it‘s right. But our teacher wants to show that proving even the simplest and easiest things can be very hard. And it is. That‘s why I have no idea how to do this.

 

[tkhunny]

"...it's basic knowlwedge..."
"...I know that it's right."

These are dangerous ideas. We see it in the political spectrum all the time. Evidence Based!!

How many ways are there to bisect a square? If you intersect a side, is that necessarily different from hitting a vertex?

BTW: What's a crooked line? I never understood how adding the word "straight", makes a line any more a line than it already was.

 

[Dr.Peterson]

I mean, it‘s basic knowledge and just by drawing this I know it‘s right. But our teacher wants to show that proving even the simplest and easiest things can be very hard. And it is. That‘s why I have no idea how to do this.

How does being very hard imply that you have no idea how to do this? What have you tried? The important thing is to start.

One thought I have is to use analytic geometry, choosing a coordinate system.

With or without that, you will need to consider cases. Where might the line intersect the square? How can you determine whether it bisects the square in each case?

BTW: What's a crooked line? I never understood how adding the word "straight", makes a line any more a line than it already was.

We do use the term "curved line" at times; or else we use the term "curve", but allow that to include straight lines! Sometimes extra words just emphasize our meaning, or prevent misunderstanding if someone comes from at it a different direction than we expect.

 

[quick_maths]

How does being very hard imply that you have no idea how to do this? What have you tried? The important thing is to start.

With or without that, you will need to consider cases.

My idea was showing/drawing all the axes of symmetry since they all go through the center point, it‘s proven. But then I noticed that every single line (every single straight line to be precise) bisects the square.

One thought I have is to use analytic geometry, choosing a coordinate system.

Yeah, I thought about that, too. But how would I calculate the areas of the square or even have a square in a coordinate system? That would be far beyond the level in our class and our teacher said, it‘s doable without higher level stuff.

 

[lev888]

Very informal:
Any line passing through the center point bisects the square (easy to prove).
Are there lines not passing through the center point bisect the square?
Let's consider an arbitrary line L not passing through the center point. There is a line C parallel to L that passes through the center point. It divides the square into 2 equal shapes. Take C and move it towards L - this makes one of the shapes larger and the other smaller. Therefore, L does not divide the square into equal shapes.

 

[Dr.Peterson]

But how would I calculate the areas of the square or even have a square in a coordinate system? That would be far beyond the level in our class and our teacher said, it‘s doable without higher level stuff.

What is the level of your class? You haven't said.

If you don't know enough about coordinate geometry, then use another approach.

But I'm not thinking of any fancy formula for area; and if you know the basics of coordinate geometry, you should be able to recognize a square. (I'd start by taking the origin at one corner, and putting one edge along the x-axis and another along the y-axis, with each side one unit long.)

Most likely, the only sense in which this would be beyond you would be that there are lots of pieces to put together. There is probably nothing you don't know that is required -- only patience.

 

[tkhunny]

We do use the term "curved line" at times;

I don't. Not everyone agrees, of course. In statistical regression we often see the tern "curvilinear", meaning not linear. It's a silly term. Always has been. Apologies if someone's grandfather invented it.

or else we use the term "curve", but allow that to include straight lines!

That's not allowance; that's generalization. Quite acceptable. When a child connects dots, one often hears that the dots are being connected with "lines". As you state, they are actually "curves", that just might be line segments.

Sometimes extra words just emphasize our meaning, or prevent misunderstanding if someone comes from at it a different direction than we expect.

Oftentimes, our attempts at preventing misunderstanding result in obfuscation.

Sadly, we tell early students of mathematics to rely on definitions, and then we accept or invent terms that ignore the definitions, or attempt to emphasize the already-clear definitions. In my personal opinion, we would do more students greater service if we abandoned this practice.

** End of Soap Box **

 

[Dr.Peterson]

Use of language in math can be quite complicated, especially when we mix together everyday usage and specialized usage. My interest is in communicating math to students, more than in communicating among mathematicians, which is a very different context.

I find it funny that one would object to the use of "line" to mean something curved (which is quite common outside of math -- just pick up a crayon to "color within the lines", or look at a "line drawing"), but not see an issue with the use of "curve" to mean something that is not curved (which is decidedly not common in everyday use)!

There are potential pitfalls everywhere, and people will have different opinions on the best way to deal with them. But I think the least we can do is to use an extra word or two here and there ("straight line") to ease newcomers' journey through this minefield. I didn't say you have to use the whole phrase, and not take "line" to imply straight -- just that it makes sense in any context where someone might possibly be unsure.

 

[quick_maths]

Yeah, those vague ideas couldn‘t help me at all, I‘ve not gotten further since I asked my question.
I expected something helpful, not some users arguing about language used in math, completely unrelated to my question.
Sorry, but I haven‘t gotten any help on here :/

 

[tkhunny]

It's never useless. Did you take a really good look at the comments from lev888? Nearly laid out the whole thing for you. Did you miss it because you thought it was just useless math arguing?

 

[Dr.Peterson]

Yeah, those vague ideas couldn‘t help me at all, I‘ve not gotten further since I asked my question.
I expected something helpful, not some users arguing about language used in math, completely unrelated to my question.
Sorry, but I haven‘t gotten any help on here :/

Sorry about the discussion going a little astray (I've seen worse). But I was hoping you'd show us some specific thoughts, and let us know what level of math you are doing, so we can help you more effectively. In particular, if you had responded to lev888's post #5 by saying whether that was or was not the sort of proof you are looking for, we'd have a better idea where to go from there.

So, if you've made no progress at all to show us, can you tell us more about the assignment? What sorts of geometry have you been learning that might be of use? Are you looking for a formal proof (the two-column sort, or a paragraph proof that similarly uses known theorems as its basis)? If so, what theorems are available to use? There are so many ways one could start, we really can't help without narrowing things down a bit.

 

[Steven G]

Very informal:
Any line passing through the center point bisects the square (easy to prove).
Are there lines not passing through the center point bisect the square?
Let's consider an arbitrary line L not passing through the center point. There is a line C parallel to L that passes through the center point. It divides the square into 2 equal shapes. Take C and move it towards L - this makes one of the shapes larger and the other smaller. Therefore, L does not divide the square into equal shapes.

Nicely done!