<font color="aqua">Can you explain this:

http://pages.infinit.net/luvian/TRIANGLE.gif

</font>

Er...someone's taken a bite out of that oddly coloured (and shaped) pie :D [img]tongue.gif[/img]

<font color="#00ccff">I have seen this before, I believe the explanation was that the slope of the red piece was not constant or some such, as in it had a slight bend. I don't really remember exactly though.</font>

But both triangle are exactly the same size, angle and all...

<font color="#00ccff">The red and green triangles are not the same size at all check your measurements.</font>

<font color="#00ccff">The slope of the Hypotenuse of the red triangle is different than that of the green, this causes the displacement when they are swapped.</font>

MagiK's right. If you take the hypotenuses of the two macro-triangles, and place them on the same diagonal you'll find that they create a disk-shaped form. This is where the "lost" area went to.

<font color="#00ccff">I hate it when I have to be all scientific [img]tongue.gif[/img]
Im the Jack O'Neile of IW [img]smile.gif[/img] </font>

Why complicate it with calculating the hypothenuses? Just check the base and height. They´re not matching. The height ratio is 1:1,5 and the base 1:1,8. There´s a methematical term for triangles that have matching ratios, but that term isn´t in my vocabulary ;) :D

<font color="#00ccff">Equilateral?

No thats matching angles....but all their sides have the same slope [img]smile.gif[/img] </font>

No it´s more basic than that. You have to differnt triangles one large and one small. The ratios between them is matching so that you can count the area of both of them by only knowing the measurements of one... No angles involved.

I just confirmed the theory by printing it out and cutting it in half and laying the drawings over top one another. There is an almost imperceptable slope change on the hypotenuse that creates a very thin convex sliver about 8 squares long - making up the missing square. Quite cool. Any teachers on this board? - show it to math students.

Yeah, the term is "similar." Your theory doesn't work mathematically though because one of the shapes is not in fact a triangle - it has a chunk cut out, and a non-linear "hyptenuse." ;)

<font color="#00ccff">You have entirely too much free time at work mister! ;) </font>

I thought you were our curmudgeon! :D [img]tongue.gif[/img]

However, people are learning. I know I did. I have been staring at that puzzle for close to nearly 5 years already at the transit stations. It feels good to know and understand the answer now. :D

Was it that simple? Oh dear lol :D ! Thanks for the info. [img]smile.gif[/img]

[ 11-07-2002, 12:37 PM: Message edited by: WillowIX ]

Actually the triangle bit is a red herring. All three triangles are similar (angles equal, sides proportional), and the total area of the reconfigured large triangle is the same in each case.

The "hole" is created by the differeces in area of the two rectangles that make the difference in the large triangle when the two small triangles are lined up.

With the small triangle on top, the resulting rectangle is 3x5, which has an area of 15.

With the med triangle on top, the resulting rectangle is 2x8, which has an area of 16.

So, when using discreate parts as shown in the picture, there is not enough material to complete the triange in the second configuration, even though the total area is the same for each large triangle.

I thought you were our curmudgeon! :D [img]tongue.gif[/img] </font>[/QUOTE]A challenging DUAL ROLE to be sure. :D

As for me having free time at work, we all obviously do. But, I've gotten several projects off my desk today, including pleadings in 2 cases, and I intend to finish up a bunch of discovery to pound the other side into the dirt by the time the day is finished. For young lawyers, a lot of the day is spent awaiting feedback on projects - I take work home at night most days.

'SNAP' *breaks arm patting self on back*

I count two triangles. The red on and the green one. The other two triangle-like forms are *not* true triangles. The first is an irregular quadrilateral - the 4th vertice is at the point where the red and green trianlges meet. The second is an irregular octagon. Your only looking at one part of the puzzle. You need to explain why the 4 pieces are the same area and in the same space, but leave a gap There are no large triangles.

Red triange is 3*8, green is 2*5. They are not similar, the gradient of the hypotenuse is different, rearranging them therefore allows for this gap to be created.

This can bee seen, if you look at the point where the triablges meet in the first picture. IF your eyesight is super human, you can see a slight angle there.

For the rest of us, look at the same point in the second diagram. The hypotenuse of the red triangle passes close to this point, not through it.

<font color="#00ccff">You know, my answer was correct and a lot less wordy than some on here [img]smile.gif[/img]

Harrumph [img]smile.gif[/img] </font>

Your sig is really going to your heads. :D Life imitates art imitates life imitates art.....

Ah, yes. I see the Illusion. It is still related to the difference in the areas of the two rectangles though. The triangles just go towards creating the illusion.

NS - the rectangles have nothing to do with it. The hypotenuse lines on the small triangles have been slightly altered. The triangles in the first drawing are simply not the same as the like-colored triangles in the second one. It's *absolutely all* in the hypotenuse lines.

The triangles are the same size in each diagram. All the components are identical. The illusion is caused by the fact that the triangles are not mathematicaly similar.

The rectagles just serve to show the effect.

Equal triangles *are* similar. Save the picture to your drive and print it out. Cut it in half and overlay the triangles. You will note everything is in the hypotenuse.

That term should be clever curmudgeon LOA - lets give MagiK some credit - he was very quick to debunk the puzzle. - (A+)

I disagree with TL when he told Willow IX that "Your theory doesn't work mathematically". Irrespecive of any curvature of the hypotenuse, he had already proven mathematically that the triangles were not similar and had differing slopes. He could have given a more complete answer though by determining all ratios :
Smallest = 5 by 2 - taking that as the basis then the next triangle that is 2 high should be 7.5 long (is 8), and the big triangle which is 5 high should be 12.5 (is 13) - so to me Willow had a mathematically valid solution. (B)

NS on the rectangles though is well off beam. (F)

TL took the inventive grade 3 solution of cutting the shapes out - hmm - how to grade this answer - I know - you get a pink elephant stamp ;)

Davros, as neither macro-shape is a triangle, and as both triangle-looking shapes actually have curved lines as their hypotenuse, there are quite technically NO triangles in the pictures. Sorry, but you don't get a gold star, either. :D Though, perhaps I should have explained myself better.

[edit - Check the slope. Run over rise of Shape #1 is almost 5/2, but not quite, that of #2 is a bit more than 5/2, and each shape has a slightly curved hypotenuse.]

[ 11-07-2002, 05:32 PM: Message edited by: Timber Loftis ]

Red = 3*8, giving a gradient of 2.33

Green = 2*5, giving a gradient of 2.5

They are *not* similar triangles, they are also not curved in any way. They will appear curved in your monitor, but this puzzle works with straight lines.

I will do the maths if I get bored tonight.

TL

I'm abstracting just a little bit more. The sides of the small triangles are exactly the dimmensions of the two rectangles - 2, 3, 5, and 8. If they were based on diminsions of the apparent large triangle, the large triangle's hypotinuse would not have intersections at (8,3) or (5,2), it would be slightly off as andrewas pointed out. So, forcing the dimensions of the triangles to be related to the two rectangles causes them to be slightly different, causing the gradients of the hypotinuse to be different, and setting up the illusion that the two large figures are triangles when in fact they are not. So taking a few levels of abstraction beyond that, the difference is the difference between the areas of a 3x5 rectangle and a 2x8 rectangle. I'm doing a visualization thing ... and I do get the math side. Even doing a similar triangles comparison, 3/8 is not proportional to 2/5. The proportion ends up being
15!=16 - which are the areas of the two rectangles. Were the proportion == then the angles would be == and the triangles would be similar, which we all agree that they are not.

And yes, Magik, you said it with much fewer words! ;)

Andrewas: Both larger shapes have a "run" of 13 squares and a "rise" of 5. Yet, we all agree the difference is found in the hyptenuse. If that line (the hypotenuse) has equivalent end-points, the only way the slope can vary is if one or both of them is curved. At least one of the lines is simply not straight.

[edit] Oh, and my statement that equal triangles *are* similar was made to disprove the assertion made by you that "all components are the same" which I took to mean you thought the triangles were equal but not similar.

[ 11-07-2002, 06:04 PM: Message edited by: Timber Loftis ]

Andrewas - thank you ....

Daveros - *bthththtlt* [img]tongue.gif[/img]

Thats right. It's not a straight line. It is actually two straight lines with an angle of apx 0 but not exactly 0. As andewas pointed out there is a vertex.

Okay, now I see where we misunderstood each other. The "curvature" I mentioned was regarding the single long hypotenuse of each shape. Now that you point out that this line is actually two lines, each of which is straight, I see we are in agreement. Thanks. PS - I don't think *any* mention of the "hypotenuse" on my part *ever* referred to the smaller red and green triangles, and that's perhaps why I was confusing folks.

Yeah, we all agree .... we were just nit picking on each others approach! And who started all of this by the way?!?! My head now hurts! Bubye! [img]tongue.gif[/img]

I agree with this statement 100%, as I've seen this geometrical puzzle before.

<font color = lightgreen>The leftmost angle in the green triangle has a measure of arctan .4 = 21.8 degrees.
The leftmost angle in the red triangle has a measure of arctan .375 = 20.56 degrees.

Take the first diagram, invert it, and slide it next to itself to create a rectangle. You would find that
there is a sliver whose area equals 1.

A wonderful problem to let your high school geometry students try to solve for a week. [img]graemlins/firedevil.gif[/img] </font>