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[Yaweee]

I just checked them, they're independent, I believe, unless I wrote something down incorrectly.

With row swaps, should be able to define a matrix with

1 1 0 0
0 1 1 0
0 0 1 1
1 0 0 1

t1 t2 t3 x

equation 1 + equation 3 - equation 2 = equation 4, so they are dependent, and some geometric trick is necessary to suss out one more equation.

 

[zoku88]

I think this is the key disconnect many of us are having. In the mindset of elementary geometry, what was stated there doesn't necessarily equate to "you can just use equations to solve for it." It's providing you rules to use for your proof as you step through it like in the example two-column proofs I posted above. "Because opposite angles are equal, I can state this unknown is n degrees." When I think of geometry class back in junior high, I don't think equations at all. I think of that style of proof we had to write up.

I still think you're confusing terms. An elementary geometric proof is merely a proof that uses the theorem available in elementary geometry. There's no specific form that they must take.

If you wanted a two-column proof (like it seems you clearly wanted,) then you should have said so in the OP (like I guess you've edited it in to include.)

But really, there's no difference in skill. In the two-column proofs, you are doing algebra anyway, it's just really dressed up.

 

[Haly]

EDIT: Spoke too soon.

 

[DanteFox]

I just checked them, they're independent, I believe, unless I wrote something down incorrectly.

With row swaps, should be able to define a matrix with

1 1 0 0
0 1 1 0
0 0 1 1
1 0 0 1

t1 t2 t3 x

when I try to reduce that I get:

1001
010-1
0011
0011

or some variation.

 

[zoku88]

when I try to reduce that I get:

1001
010-1
0011
0011

or some variation.

Oh wow, you're right.

I'm weak when it comes to combining rows (or numbers in general.)

 

[DanteFox]

I think I solved it without matrices though. I'll post it up.

 

[XiaNaphryz]

I still think you're confusing terms. An elementary geometric proof is merely a proof that uses the theorem available in elementary geometry. There's no specific form that they must take.

If you wanted a two-column proof (like it seems you clearly wanted,) then you should have said so in the OP (like I guess you've edited it in to include.)

But really, there's no difference in skill. In the two-column proofs, you are doing algebra anyway, it's just really dressed up.

Hell, the original problem even stated you don't need to use a two-column proof, and that you can just describe it in a paragraph. Maybe the author was a bit ambiguous to some of you, but it seemed clear to me what style proof the problem was looking for. Just quickly scrolling through the answer that was posted also confirms it to me as well.

 

[zoku88]

Hell, the original problem even stated you don't need to use a two-column proof, and that you can just describe it in a paragraph. Maybe the author was a bit ambiguous to some of you, but it seemed clear to me what style proof the problem was looking for. Just quickly scrolling through the answer that was posted also confirms it to me as well.

I'm not sure the style is important. I really think he just wanted people to use elementary geometry (thus the title.) Why would someone care about style anyway? He never even stated what type of proof to use (which is why we were using algebra.) He gave tips on how to write a proof in general, though...

 

[DanteFox]

nm I thought I had it but I messed up.

 

[Hari Seldon]

So we aren't just supposed to use x+y+z = 180 + algebra?

 

[Haly]

I'm not sure what's up but...

As far as I can tell, X can be any number from 1-129

 

[XiaNaphryz]

I'm not sure the style is important. I really think he just wanted people to use elementary geometry (thus the title.)

Exactly. Just elementary geometry. No trig, no algebra outside the basic equations given, no matrices, nothing else beyond what is stated in the "Here is everything you need to know to solve the above problems" section.

 

[zoku88]

Exactly. Just elementary geometry. No trig, no algebra outside the basic equations given, no matrices, nothing else.

Uhm, we ARE using the basic equations given. The matrices were only a compact way of representing the equation given, so they clearly don't break the rules given by the problem poster. Everything done so far is clearly in the realms of elementary geometry.

 

[XiaNaphryz]

Uhm, we ARE using the basic equations given. The matrices were only a compact way of representing the equation given, so they clearly don't break the rules given by the problem poster. Everything done so far is clearly in the realms of elementary geometry.

Okay, now you're just fucking around with me.

 

[zoku88]

Okay, now you're just fucking around with me.

I'm being quite serious.

using a matrix to represent the system of equations is no different from differing the text from english to jpn. As long we don't use any techniques that you would use in linear algebra class, it would be fine. (as in, we would probably have to solve the equations one by one.)

 

[Timedog]

Uhm, we ARE using the basic equations given. The matrices were only a compact way of representing the equation given, so they clearly don't break the rules given by the problem poster. Everything done so far is clearly in the realms of elementary geometry.

Dude, seriously, come off it. did you learn what the a matrix was in geometry class in middle school/early high school? Were you solving systems of equations in geometry class? Oh, you weren't? Then quit arguing about stupid semantics and do the problem as it is intended. You know exactly what he is talking about, and you know exactly how the problem is intended to be solved. I mean, or you could just continue on your incessant quest to be obnoxious. If it's outside the bounds of a geometry class textbook, don't use it. It's an incredibly simple concept.

 

[zoku88]

Dude, seriously, come off it. did you learn what the a matrix was in geometry class in middle school/early high school? Oh, you didn't? Then quit arguing about stupid semantics and do the problem as it is intended. You know exactly what he is talking about, and you know exactly how the problem is intended to be solved. I mean, or you could just continue on your incessant quest to be obnoxious. If it's outside the bounds of a geometry class textbook, don't use it. It's an incredibly simple concept.

I never said we could use matrix algebra (i said that we clearly couldn't,) but we could clearly use basic algebra in geometry class, so I don't know what you're talking about... We're solving problems using basic algebra and basic geometry, which is totally inside what was said in the first two lines of the problem. I'm not sure what your problem is.

Use geometry to find equations... solving equations one-by-one. How does this NOT fit the problem? Geez, we only used the matrix to find out if our set of equations was solvable...

 

[XiaNaphryz]

I'm being quite serious.

using a matrix to represent the system of equations is no different from differing the text from english to jpn. As long we don't use any techniques that you would use in linear algebra class, it would be fine. (as in, we would probably have to solve the equations one by one.)

The problem presented was one such that someone who had just learned elementary geometry could understand the proof. Said person is likely not to have learned anything beyond algebra, including basic trig, matrices, or calc. It's quite clear what type of solution is being requested here.

 

[Hari Seldon]

x=67.5 ?

 

[bachikarn]

You can use matrices. It's not a big deal. Unless the answer is that two of the triangles are congruent, and then x is equal to one of the angles already on the diagram, you are going to have to use SOME algebra to solve it. Using matrices are just a easy and systematic way to solve the algebra. You can use normal techniques if you want, but we aren't solving the problem using crazy linear algebra.

 

[Haly]

Working out the proof bit by bit, will have it in a moment. Tricky bastard.

 

[XiaNaphryz]

You can use matrices. It's not a big deal. Unless the answer is that two of the triangles are congruent, and then x is equal to one of the angles already on the diagram, you are going to have to use SOME algebra to solve it.

It's been already stated that you don't need to. The only equations needed are "the sum of adjacent angles is 180 degrees" and "the sum of the interior angles of a triangle is 180 degrees." So if you don't need to do any algebra to solve the problem, you shouldn't have to use matrices either.

 

[ninjarr]

Not bad, not bad. A fun problem.

Here is another for anyone who is bored. While a bit easier, it has a far cooler result.

Can two triangles share 5 of 6 properties in common (3 side lengths, 3 angles) and not be congruent?

Hint:

Spoiler

Haha yeah right, get back to work slacker.

 

[Haly]

Proof is WRONG.

Spoiler

The intersection in the middle will be vertex F

1. Angle AFB = 50
P: Sum of angles of triangle is 180

2. Angle EDB = 130
P: Triangle of straight line is 180

3. Angle EDB = Angle EDB
P: I forgot what law this was called

4. Angle DEA = Angle CBD
P: 20 = 20

5. Triangle EDB is similar to triangle FDE
P: Angle-Angle (4, 5)

6: Angle DEA = Angle CBD
P: Corresponding angles of similar triangles are equal

7: Angle DEA = 20
P: (6)

Since the challenge was in finding the proof, not the answer I assumed x in order to tease out the process. Cheating I guess, but it works. Whoever made this question was a clever asshole.

My 8th grade teacher would be ashamed.

 

[DanteFox]

Spoiler

The intersection in the middle will be vertex F

1. Angle AFB = 50
P: Sum of angles of triangle is 180

2. Angle EDB = 130
P: Triangle of straight line is 180

3. Angle EDB = Angle EDB
P: I forgot what law this was called

4. Angle DEA = Angle CBD
P: 20 = 20

5. Triangle EDB is similar to triangle FDE
P: Angle-Angle (4, 5)

6: Angle DEA = Angle CBD
P: Corresponding angles of similar triangles are equal

7: Angle DEA = 20
P: (6)

Since the challenge was in finding the proof, not the answer I assumed x in order to tease out the process. Cheating I guess, but it works. Whoever made this question was a clever asshole.

My 8th grade teacher would be ashamed.

lolwut. that's a terrible proof. Some of that didn't even make sense. I'm calling BS on this one.

 

[Haly]

lolwut. that's a terrible proof. Some of that didn't even make sense. I'm calling BS on this one.

I admit I haven't done a geometry proof since forever so I might've messed something up.

Oh woops, I see what's wrong. Crap, well that messes everything up.

 

[nincompoop]

ITT: "This is so easy! It's completely obvious that the answer is (wrong answer)."

 

[rage1973]

It's much easier when you draw some parallel lines and make a few parallelogram in the triangle.

 

[MightyHealthy]

my head hurts

 

[Timedog]

It's much easier when you draw some parallel lines and make a few parallelogram in the triangle.

fuck, i was drawing squares in the triangle, but parallelograms might make more sense.

 

[ElectricBlue187]

This is so easy, why are there 2000 words in the OP.

exactly

 

[Timedog]

exactly

ITT: guys who are geniuses. on the internet.

 

[watervengeance]

The proof from sun.com just blew my mind. And I thought I was good at math.....

 

[Yaweee]

So, the "t1 + t2 + t3 =180" sort of rules looked like they would work, but it doesn't because you surprisingly end up with linearly dependent equations.

My question is this:

For geometric problems like this one, are there any general rules that would state when you'd get linearly independent equations, and when you wouldn't? It genuinely seemed like you were extracting discrete pieces of information, but it's degenerate.

 

[ProfessorLobo]

ITT: "This is so easy! It's completely obvious that the answer is (wrong answer)."

Actually, most people said it was easy without giving an answer. Now we think that they're good at math, and they don't have to prove it.

 

[Timedog]

Actually, most people said it was easy without giving an answer. Now we think that they're good at math, and they don't have to prove it.

The internet, where everyone thinks they're a world reknowned rocket scientist, cosmologist, linguist, biologist, famous writer, comedian, and philospher rolled into one.

 

[FFChris]

Do half of you ever bother to read the OP, or do you just post the first thing that comes to your head?

 

[XiaNaphryz]

So, the "t1 + t2 + t3 =180" sort of rules looked like they would work, but it doesn't because you surprisingly end up with linearly dependent equations.

My question is this:

For geometric problems like this one, are there any general rules that would state when you'd get linearly independent equations, and when you wouldn't? It genuinely seemed like you were extracting discrete pieces of information, but it's degenerate.

I haven't had to do any linear algebra since college over 9-10 years ago, so I can't for the life of me remember if there was a general way to recognize if a geometric situation was linearly dependent or independent. I vaguely recall something about a set of vectors being linearly independent if you can prove that the only representations of the null vector are trivial?

Do half of you ever bother to read the OP, or do you just post the first thing that comes to your head?

 

[Feep]

Here's a fun question. This took me six days to complete.

Two circles of radius r exist on a plane. How far apart should the centers of these two circles be such that the area of the overlapping region of the two circles is equal to the combined areas of the non-overlapping regions?

Edit: I just looked up the solution to the OP, because I fucking hate geometry. The solution is quite difficult and not at all elegant, in my opinion. Though, to be fair, neither is the answer to my problem. = D

No, the answer is not 67.5 = P

 

[Hari Seldon]

So, the "t1 + t2 + t3 =180" sort of rules looked like they would work, but it doesn't because you surprisingly end up with linearly dependent equations.

My question is this:

For geometric problems like this one, are there any general rules that would state when you'd get linearly independent equations, and when you wouldn't? It genuinely seemed like you were extracting discrete pieces of information, but it's degenerate.

You can still solve it, you just need some more advanced linear algebra. That is why I asked if 67.5 is the answer, because that is what I got by using linear algebra.

 

[Javaman]

I love how a good chunk of people are posting what a piece of cake it was without even working all the way through the problem. It was smooth sailing for the first 5 or so missing angles but it all comes screeching to a halt around D.

 

[Haly]

I love how a good chunk of people are posting what a piece of cake it was without even working all the way through the problem. It was smooth sailing for the first 5 or so missing angles but it all comes screeching to a halt around D.

System of equations won't work out, the solution is probably a very complicated series of additional parallel lines and application of the angle/triangle rules.

I'll solve this eventually.

 

[Earthstrike]

Let the intersection of lines AE and DB be F.
AFB = 50 (sum of internal angles of a triangle.)
DFE = 50 (opposite angle rules)
EFB = 130 (adjacent angles of a bisected line sum up to 180)
DFA = 130 (adjacent angles of a bisected line sum up to 180)
DCE = 20 (sum of interior angles of a triangle)
FEB = 30 (sum of interior angles of a triangle)
ADF = 40 (sum of interior angles of a triangle)
CED + DEF + FEB = 180
CED + DEF = 150
CDE + EDF + ADF = 180
CDE + EDF = 140
CDE + CED + DCE = 180
CDE + CED = 160
EDF + DEF + DFE = 180
EDF + DEF = 130
CDE + CED + CDE + EDF +CED + DEF + EDF + DEF = 580
CDE + CED + EDF + DEF = 290
Given the system of 4 equations in 4 unknowns, the system is not exactly solvable and requires an additional piece of information.

I'm stuck here, I believe you have construct some kind of line out of the sides of the triangle so that congruency implies an angle.

 

[cashman]

"elementary"? WTF I just learned this last year!

 

[Gilgamesh]

Edit: Hold on a minute...

 

[Aegus]

Err am I being dumb or the question basically asking how I came to define x in English?

In a question like this I'd provide both an equation answer and a proof stating all the steps and logic used in the equation.

 

[loosus]

So I'm stuck after I did my "the sum of all angles of a triangle equals 180 degrees" crap.

And I don't know where to go from here. I am guessing that congruency has something to do with it, but I am having trouble seeing how you can conclusively say that anything is congruent.

 

[ImNotLikeThem]

Am I the only one that got 50?

 

[Saargabath]

wait a sec XD

 

[XiaNaphryz]

Err am I being dumb or the question basically asking how I came to define x in English?

It's asking to solve for X using only what's stated in the "Here is everything you need to know to solve the problem" section.