Triangle paradox

Edgardo

Check out this riddle:
http://www.mathematik.uni-bielefeld....w-paradox.html

Where does the hole in the lower figure come from?

lawtonfogle

when rearanged the triangle area, counting the square, is bigger. I'll try to explane this by a step by step post. The bottom left of the graph be the origin.



notice all we are adding is just 1 square unit. 1 is not that big and when split and added to 25 other units, it can be hard to notice the change, expesily if grouped together.

( u might be like this guy right now so let me cleat it up with an example)

an example is,

a 10 by 10 square on the graph is increased in size so that instead of 100 area it has 101 area, the change can be hard to notice, very hard if you dont know what to look for.

( i hope your like this guy who understands it now)

lawtonfogle

(pull up the picture and make it small, it will be helpful)

look at (6,11), there is a little space to the SE of it

now look at (6,3), that space is not there . actualy, there is some space up to the NW. one might say the figures are not drawn to scale, yet if they were it would be the same problem. This is the 1 area that is split and added to all the edges. The area, if measured from a triangle drawn to scale, would be 1 bigger for the lower triangle than the upper one, if you measured 'EXACTLY'.

RandallB

Not from that "Easy" Explaination? Can't be true because as the prolem states all four parts are the same!

To show Edgardo you "get it" tell him the NAME of the source for the "extra" space in one word - just one.

Hint in white:
The word is a spiecal case for the correct name of the two "Large triangles"

Telos

I've never heard of "Fibonacci Bamboozlement," but that first "triangle" is actually a quadrilateral. There's a barely perceptible "dent" in the first figure and a "bump" in the second.

You can tell because the slopes of the two smaller triangles are different (2/5 green, 3/8 red).

Doctordick

You're right on the money. You show strong signs of being able to think!

Have fun -- Dick

RandallB

So what do you call the 2nd "triangle"??
And back to the original question - where did the "extra" space come from??
You should be able to define it in just one word.

Keep thinking you'll get it.

Edgardo

I would day that Telos gave a good answer (I had the same thought).

RandallB

I assume a good answer is good enough for you or you haven't figured it out yet. Another hint - cover the "Large triangle" with the "small triangle" and trim off any exposed part of the large.
Now discribe the triming.
And no " quadrilateral " will not do.

Telos

Randall, the "hole" comes from a rearranging of the shapes.

There is no lost area, so there is no hole.

RandallB

What you think the triangles are the same size???
Obviously you didn’t do what I’d suggested - so why are you commenting? Use a real pair of scissors cut out four real “shapes” and trace out two “triangles”. Then really trim out the “extra” space on the larger using real scissors and you can actually have it in your hand. Then get back with us and tell us what it is.

Telos

Randall, when you said, "large triangle" and "small triangle" I thought you actually meant the real green and red triangles - because, neither of the two larger figures are triangles. So your statement was very confusing. I meant no offense by not responding to it. Moreover, this is a math puzzle and this is a math forum. And we're not kindergarteners. We should be able to figure this out without resolving to construction paper.

green triangle area => (1/2)2*5 = 5 units
red triangle area => (1/2)3*8 = 12 units
yellow polygon area => 7 units
light green polygon area => 8 units

total area for both figures => 5+12+7+8 = 32 units

The area is the same for both figures. There is no lost area. There is no missing section.

Now, if I'm making a mistake, please tell me. And stop keeping us in suspense, let us know the word you're thinking of. Is it "tangram?"

RandallB

There in (" ") because as you know there are NOT triangles! Of cource you shouldn't but sometimes using somethink real helps get your abstract mind to work better!
NO they are NOT the same -- If needed turn the rectangle over created by the Polygon’s so the space is in the middle of the shape!!! The larger one is larger! Why? one word will do!First get you math right, if you say you cut it out and still don't "get it" then sure I'll tell - but when you do get it you'll look back on this thread to find a massive number of hints. Working brain teasers is about learning from the doing not having answers given to you!

Telos

"Tautology?"

You want me to cut out triangles and squares to help my abstract mind work better? These are triangles and squares. You're insulting me and wasting my time.

You've just been added to my ignore list.

Uno Lee

It took me a while but here is what I got Large triangle slope =3:8
Small triangle slope=2:5
lowest common denominator = 40 Large triangle slope 15:40
Small triangle slope 16:40
the slopes are different: we assume the complete structure is a triangle but it isn't. It is actually a four sided object, including the "missing square." Another way to put it is the space taken up block in the basement is made up for by flipping the roof so the water doesn't pool. You got to look at this whole thing on a bit of an angle.

RandallB

Pretty good Uno Lee at least your thinking, as you’ll note earlier in the thread the word you’re looking for to describe the “triangle” is quadrilateral. And the extra area inside the larger is also a quadrilateral! A special case quadrilateral called a square! So the question remains where did the area displaced by the square go to in the larger quadrilateral? (trust me this is simple stuff) As also hinted at earlier it shouldn’t be a surprise that the answer is a quadrilateral, but that would the best description of it. (Just like no one calls a square a quadrilateral even though it is.)

The thing I like about this puzzle at it so many people that “know so much” quickly get this and move on from it without completely “getting it”. When they do they would immediately know this word and why it’s important, and how it fits so well with what you already know. Then you’ll know you “completely get it”.

This process of learning how you yourself learn and discover, within yourself is a valuable thing and I won’t rob you of it by just telling you the answer. Trust me developing that kind of personal skill will help you in the future with ideas like Relativity, Quantum…, or whatever without having to depend on others to think for you.

But, if you really want to give up the easy way to find the answer since this Forum is full of old stuff like this! Just “Search This Forum” under brain teasers for triangle.
It's pretty simple and since I know the answer I’ll let you post it in this thread. Just let us know if you looked it up or figured it out.

AC

similar triangles
the 2 triangles are supposed to be similar.
but 2,5 and 3,7 small squares are not valid