http://205.252.89.156/triangle/Triangle.gif
This is driving me nuts and I can’t figure out why this works. Please help.
Thanks.
This is an old puxxle in new dress. It’s a scam.
Please note that the RED triangle is 8 units wide by 3 units high. The GREEN triangle is 5 units wide by 2 high. The angles in these triangles are different – they’re not similar triangles. So when you assemble them with the two odd-shaped pieces you will not, in either construction, get one supertriangle of the same shape – you’ve got an irregular quadrilateral. More accurately, you’ve got two different quadrilaterals, depending on how you assemble them. There’s no reason to expect them to have the same area, since they’re not the same shape. And they don’t. So one can have a “hole” in it.
Because the two triangles don’t meet exactly at the coordinate (8,3); it’s more like (8,3.077). And the dark green triangle isn’t one at all, but a parallelogram with a very small left side.
Just a minute, and I’ll work out the math.
Well, the area of the top triangle is 13*5/2 = 32.5.
The area of the bottom triangle is 32.5 - 1 = 31.5.
Now, in geometry, it is true that if you move stuff around so that the area (or volume) decreases, the perimeter (or surface area) increases.
The perimeter of the top triangle is
13 + 5 + sqrt((13^2)+(5^2)) = 31.92.
The perimeter of the bottom triangle is:
5 + 7 + 1 + 1 + 1 + 5 + sqrt((13^2)+(5^2)) = 33.92.
So nothing nefarious is happening here. As for “where” the hole comes from, well, it just happens that the triangle is divided up into shapes that, when moved around, conveniently make a congruent trianlge with a piece missing.
CalMeacham has it right, of course. To illustrate the point though, look at the top drawing, and check out the point where the ‘hypotenuse’ crosses the second-from-bottom grid line. It isn’t right at a grid intersection, like it is in the bottom drawing. You can see the same problem on the third-from-bottom grid line in the bottom drawing.
If the ‘hypotenuse’ was actually straight, it would cross the grid at the same points. Therefore, there is a non-straight angle at the point that the two triangles touch.
This was asked as well in MPSIMS, and I threw out my answer there, which was basically what Cal Meacham said. (What looks like a straight line in the angle on the triangle is actually two differently angled lines; on the top triangle, they’re angled concave, on the bottom, convex; thus the ‘difference’ in area is just making up for the extra area covered by the differences between the positions of the lines.)
Simetra- if you still don’t understand what we mean, draw it for yourself at four or five times the scale. You’ll notice that the “full” shape isn’t really a triangle at all; it just looks that way because it’s such a small difference. But small differences over great lengths do wreak havoc upon the surveyors, or something like that.
Thank you all very much. 
On a side note, I feel really dumb for not thinking about slope when I looked at this. ** Doh! **
Thanks again tho’. I knew there was a simple answer that i couldn’t wrap my tiny little brain around.
CalMeacham’s explanation is good. If you look VERY closely at the .gif, you will see that the “hypotenuse” of the big “triangle” is bent up in one case and down in the other. If they had wanted to make the illusion better, they could have not used the “graph paper” background which makes the difference more visible. Compare the vertical line where the red and green triangle’s hypotenuses (hypotenii? hypotenum?) meet in one picture to the other picture.
OK, here goes:
[ul]
[li]Red triangle has cornerpoints (0,0), (8,0), and (8, 3[sup]1[/sup]/[sub]13[/sub]); area of (8 * 3[sup]1[/sup]/[sub]13[/sub])/2 = 12[sup]4[/sup]/[sub]13[/sub]. But using visual cues of grid, it looks like the area should be (8 * 3)/2 = 12[sup]1[/sup]/[sub]2[/sub] square units.[/li][li]Green “triangle” is really a parallelogram with a left side of [sup]1[/sup]/[sub]13[/sub]. So its area is 5 * (2 + [sup]1[/sup]/[sub]13[/sub])/2 = 5[sup]5[/sup]/[sub]26[/sub]. Using grid cues, however, it looks like it has an area of (5 * 2)/2 = 5 square units.[/li][li]Orange area is 7 square units; light green area is 8 square units.[/li][/ul]
Adding these: 12[sup]4[/sup]/[sub]13[/sub] + 5[sup]5[/sup]/[sub]26[/sub] + 7 + 8 = 32[sup]1[/sup]/[sub]2[/sub], which is the area of the original triagle before piecing it up.
Now, rearranging them, the red triangle, orange “gun” and dark green “triangle” meet at (5,2). But the two “triangles’” diagonals don’t go to where they appear. The red one’s top point is now at (13, 5[sup]1[/sup]/[sub]13[/sub]), not (13,5) as with the original large triangle. Similarly, the dark green “triangle” doesn’t come to a point at (0,0), but its diagonal goes to (0, [sup]1[/sup]/[sub]13[/sub]).
If you drew a line from (0,0) to (13,5), you’d see a tiny sliver of each “triangle” is above the line. These sliver areas add up to exactly one, which is the missing area between (7,0), (7,1), (8,1), and (8,0).
Look at the point where the red and green triangles meet in the upper arrangement. That point is inside the area in the lower arrangement.
Now look at the point where the red and green triangles meet in the lower figure. That point is outside the area in the upper figure. The “hypothenuse” of the figure has moved up and made space for the extra square.
Dang, I thunk that extra square had been confiscated by the U.S. government and hidden in Area 53 outside Roswell, as part of the great cover-up… because that missing square held the 18 minutes of tape that would have proved that Nessie shot JFK with a weapon taken from the alien spaceship that the government has been covering up.
Similar puzzle recently posed by James Randi can be found at the bottom of this page.
His solution is on the next week’s column.
[QUOTE]
*Originally posted by AWB *
**
[li]Green “triangle” is really a parallelogram with a left side of [sup]1[/sup]/[sub]13[/sub].**[/li][/QUOTE]
No, the dark green triangle is really a triangle. The fully assembled shapes are not triangles, just as Cal and others have said. Also, if the dark green shape did have a left side of [sup]1[/sup]/[sub]13[/sub], that would make it a trapezoid, not a parallelogram.
Nice job making that fraction appear though.
[QUOTE]
*Originally posted by Greg Charles *
**
Darn it all. I did mean trapezoid.
But depending on how you cut it, either the dark green or orange areas are larger than they appear. See details around the point (8,3):
Green area is trapezoid
Green area is triangle
Either way, the reassembled triangle is no triangle.
Thanx about the fractions. I didn’t want to resort to using the vB “code” markers.
Yet another example of how the eye deceives.
The red and green pieces are both triangles but they are not congruent (identical angles, proportional sides)
If you look at the overall ‘triangle’ (it really isn’t a triangle) the upper one has a slightly concave hypoteneuse while the lower has a slightly convex hypoteneuse. The area bounded by the concave and the convex edges is equal to 1 square, hence the increase in area of 1 square in the second arrangement.
The angle of the left hand corner of the green triangle is 21.80 deg, the red triangle: 20.55 deg.
Now for some mathematical proof:
The areas of the shapes are:
Green triangle: 5 (52)/2
Red triangle : 12 (83)/2
Orange shape : 7
Lime shape : 8
Sum : 32
Now if we consider the ‘theoretical’ triangle defined by the three extreme corners of the arrangement, it’s area is:
(13*5)/2 = 32.5
The difference between these two areas is 0.5 which gives a total difference of 1 between the two arrangements.
It can also be proven with trigonometry if needed.
Just to clarify on my previous post,
A straight line passing through pts (0,0) and (13,5) will not pass through pt (5,2) or (8,3)