Can someone explain this??

[ModelM]

http://www.simeonmagic.com/triangle/triangle1.htm

 

[CTho9305]

look VERY VERY carefully at the hypotenuse. VERY VERY VERY carefully. look at the slopes of the 2 triangles that make it up

 

[Crankydog]

In other words, those two small triangles aren't exactly the same. They don't have the same small angles.

On closer inspection you'll realize that the assembled top triangle ain't no real triangle as the hypothenuse is really two lines with a very slight angle between them that makes a small bump.

The lower assembled "triangle", with the two small triangles now reversed, has the angle reversed and makes the big "hypothenuse" dip slightly towards the inside.

When you do the numbers, you'll see that numbers come out right and that small empty square is quite allowed to exist where it is.

 

[wellerdball]

http://www.simeonmagic.com/triangle/triangle1.htm

 

[kuk]

I've drawn the same triangle here, eliminating the border and using an error-proof, copy-and-paste method.

Here's the result.

First, you should notice that the orange triangle isn't really perfect. This leads to a different placement in the second triangle. The absent square is 400 square-pixels, the same exact size of the 4x100 extra border at the right of the second triangle.

Hope this helps ...

 

[Peter]

No sorry, this is wrong. There is no cheating involved.

Let's look at the surfaces.

Blue triangle: (5x2)/2 squares.
Orange triangle: (8x3)/2 squares.
Green thing: 7
Yellow thing: 8

That sums up to 5+12+7+8 = 32 squares.

If the entire thing were a triangle, the covered surface would be (13x5)/2 = 32.5 squares. So now we know the thing we look at is not a triangle to start with.

So how come it isn't? Simply because what looks like the hypotenusis isn't a straight line: The blue and orange triangles do not have the same incline - 2/5 is more than 3/8, so the assembled thing actually is slightly concave.

Assemble the pieces the other way round, and you get a slightly convex thing, with the long thin difference between the concave and the convex assembly amounting to exactly one square.

regards, Peter

 

[kuk]

I have overlaped the orange and blue triangles, and the angles are the same. To compose the smaller "triangles", I used the same straight line (off course, with anti-aliasing disabled), so there is no way that their angles would be different. If you compare both images, pixel-per-pixel, you'll notice that up until the 13th square (from left to right), the triangles are the same (of course, disconsidering the "hole" in the second).

The problem really comes from cheating on the original image. Using thick, anti-aliased lines, the orange "triangle's" imperfection is masked. And as I've tried to point out, the orange shape is not a triangle. Because of this, you cannot apply a simple tangencial formula to determine the angle of the slope. That's why I consider that the original image cheats, leading the reader to think that both smaller triangles are really triangles, which they are not.

 

[RossGr]

I believe that each of the small triangles is just fine, but the red and green triangles are not similar (in the formal since). If you count squares and compute the tangent of each to find the actual value of the acute angle you will see this. Euclidean Geometry tells us that for the large triangles to be congruent the red and green triangles must be similar, they are not.

for the Red we have
Tan = O/A = 3/8 => ~20.5deg
For the green we have
Tan = O/A = 2/4= 1/2=>~26.5deg

so while each of the small triangles are fine, neither of the larger composition shapes are even triangels. In reality they are quadralaterals with one angle nearly 180 deg.

 

[Crankydog]

Kuk, not to trash your picture or anything, but forget about precisely the orignal picture is drawn with the thick border or anything. Just count how many squares high and long all shapes are.

The red and dark green triangles are really triangles of 3x8 and 2x5 dimensions. That makes the two smaller angles 20.55 and 21.8 degrees. With all the pieces assembled, the top assembled "triangle" should be 5x13 squares wich gives an angle of 21 degrees. Depending on how you place the two smaller triangles, you'll get a slightly concave or convexe "hypothenuse". This small difference in assembled surface area is exactly equal to one square!

Both small triangles are true triangles, it's once assembles that you have the illusion of a triangle.

 

[kuk]

Originally posted by: Crankydog
Kuk, not to trash your picture or anything, but forget about precisely the orignal picture is drawn with the thick border or anything. Just count how many squares high and long all shapes are.

The red and dark green triangles are really triangles of 3x8 and 2x5 dimensions. That makes the two smaller angles 20.55 and 21.8 degrees. With all the pieces assembled, the top assembled "triangle" should be 5x13 squares wich gives an angle of 21 degrees. Depending on how you place the two smaller triangles, you'll get a slightly concave or convexe "hypothenuse". This small difference in assembled surface area is exactly equal to one square!

Both small triangles are true triangles, it's once assembles that you have the illusion of a triangle.

Yes, mathematically, I understand why these aren't really triangles.
What I'm trying to do here is purify the original image so one can understand how the spoof is done. You can do this in two ways. One is what you guys are saying, by considering the two smaller triangles TRUE triangles (8x3 and 5x2), and the final composition a quadrilateral, or a triangle with a curved hypothenuse. Or you can consider the composition a true triangle, and consequently the smaller shapes false triangles ... which is what I have drawn.

We are both arguing about the same true results ...

Kuk

 

[Peter]

No sorry kuk, it's all in the math and nothing in slightly off-measure pieces. You are wrong, math is right. Period. No need to argue. 3/8 does not equal 2/5, that's all that's to it.

 

[kuk]

Originally posted by: Peter
No sorry kuk, it's all in the math and nothing in slightly off-measure pieces. You are wrong, math is right. Period. No need to argue. 3/8 does not equal 2/5, that's all that's to it.

Oh god Peter! Hehe ...
I'm not arguing that the math is wrong! What I've said is that there is two ways to approach this problem. If you read my previous post, you'll notice that I said that I agree with the resolution presented, but I've shown a different way to see this. Since the original image can lead to two different interpretations, everyone decided to consider the smaller triangles as true triangles, whereas I approached as the big triangle composition as a true triangle, and then discovered the imperfections of the smaller shapes.

Using my approach, I've given visual proof as to where the piece bitten off went ... the 20x20 square became a 100 x 4 rectangle. It might be easier to SEE this than if tried to count, pixel per pixel, the difference in area between a perfect composition and a "3/8+2/5" combined composition.

Why be so stuborn?
Sometimes I think that people miss the point believing that there is only one answer to the world's problem. Multilateralism > unilateralism.

Kuk

 

[Peter]

Drawings are are inherently imperfect, they're made to illustrate an exact (or geometrically perfect if you please) problem. Any reasoning built on the imperfection of the illustration is just plain missing the point.