Okay ************* ...
Lets call the smallest angle in the green triangle A:
Before rearranging Tan A = 3/8
Making A 20.566 Degrees
After rearranging Tan A = 5/13
Making A 21.038 Degrees
Therefore we can see that there are minor changes to the angles, creating the extra square.
I'm sitting here in the laundry room in college, having a few drinks, doing some laundry (There's nothing else to do in here) and this post was going to drive me ******* crazy. Thank you a million times for posting this. I just might love you.
It's simple, red and green have slopes of 3/8 while blue and orange have a slope 2/5.
Thus the lines are not exactly straight. there is a small gap in between the blue and the green piece, and the red and the orange piece. That gap creates the extra 1 area.
the only reason this "works" is because it looks like it works... in fact, the slopes of the two hypotenuses of each right triangle are just slightly different. in the actual triangle and the triangle in the polygon, it can be found by finding the hypotenuse length (8^2 + 3^2 = 8.54) and (5^2 + 2^2 = 5.39), then you find the angles where they all meet and find that out by using sin^-1 cos^-1 or tan^-1, to see that their angles are really 68.2 degrees and 69.5 degrees, a very slight difference, so it is not blatently obvious and does not really mean anything
tl;dr, angles that are touching are really 68.2 and 69.5 and aren't the same so the above doesn't work
its not about the angles, its the area of the square, you need not do anything more than count the little squares, an 1 square comes out of nowhere in the second construct
But that is a line, it does not exist, it is implied that it isn't really there and you are just supposed to push the 2 bottom pieces to the left to close the space, in which you would remove the gap, and still have the same answer as given
Because the two triangles have different angles, they don't fit perfectly together, while the lines are in fact implied, and they don't actually have length, the space between the expressed lines is equal to one square, but you can't see the space since the line is covering it up.
After staring at for five minutes, i realized that problem was coming from the difference in slopes between the two types of shapes. the ones on the top have diagonal slopes 3/8(or 8/3, depending on what direction their in), while the bottom ones have a diagonal slope of 5/3(3/5). This means the diagonal don't actually line up properly, meaning the second shape wouldn't be a true rectangle.
to prove this, take the 4 seperate shapes and find their area, then add them together. 1&2 both have areas of (8*3)/2, which is 12 each. pieces 3&4 are a bit tougher, because their rectangles and triangle put together, but you can calculate them as (3*5)+[(2*5)/2], which comes out to 20 each. put it all together (12+12+20+20), and you get 64. This is obvious proof that the "15x5" rectangle or it's pieces have been distorted or misplaced in some fashion.
If you were to enlarge the scale of this to something much more... colossal, you'd notice that the slopes don't match up.
At a glance, no matter how long, it seems to match, but that's because the scale is 8x8 and the margin is just a singular unit spread out to the point it's impossible to notice.
Right, because the slope of 3/8 (the division of the red and green triangles) is not equal to a slope of 2/5 (or 5/2, when you rotate it 90 degrees, the division of the orange and blue trapezoids)
Mhm, but because 3/8 and 2/5 are so close to each other, as well as being put in as small a size as a simple gif, it's incredibly difficult to tell the difference without any form of check-up, so most people assume the gif was correct.
Not exactly sure what you meant by that, but basically the lines aren't actually straight so by fudging it a bit, you can get it to look like its one unit^2 bigger even though its not