The magic triangle problem!!!?

Question

Can anyone explain this -
http://www.cs.vu.nl/~mathijs/brainteasers/magictriangle.gif

Answer 1

The answer lies in the fact that neither shape is actually a triangle, they are quadrilaterals. They have four sides.

The side that looks like it is the hypotenuse is actually two sides, the corner being where the green and red triangles meet.

The angle of this corner is very close to 180 degrees, which makes it look very similar to a straight line, but it isn't quite 180.

You can see this by looking on the first shape, and the lower left corner of the green triangle in the upper image. You can see that it is exactly on the intersection of the gridlines. However, when you have a look at the same position on the lower image, you can see that the side of the shape passes quite a way to the left of the gridline intersection, rather than directly over the gridline intersection as in the first image.

This has the effect of basically bending that side of the shape in or out. In the upper figure, the sides has been bent in. In the lower figure, the side has been bent out. By bending it out, it has increased the area of the shape by 1 square. because the two "triangles" are made up of the same shapes, the total area cannot change, thus the extra square of area added by bending the side out must be removed from another area of the shape, which accounts for the empty square along the bottom.

Answer 2

The snag is that these four shapes seem to form a triangle together, but it isn't a real triangle! Compute the areas of the shapes:

Yellow: 7
Green: 8
Red: 12
Blue: 5
Total: 32


Together they seem to form a triangle with an area of (5 * 13) / 2 = 32.5

In the first case, the shapes together form an area of 32, which is 0.5 less than the area of triangle they seem to form. In the second case, they form an area of 33, which is 0.5 more. So the difference in area is 1, and that's the "hole" in the second area

The tricky part is the hypotenuse of the "virtual triangle". The steepness of the hypotenuse of the red triangle is 3 / 8 (= 15/40), that of the blue one is 2 / 5 (= 16/40). So the one of the blue triangle is a very little bit steeper. You almost don't see it, especially if you draw them on a grid...or can you see it now?

Answer 3

"Magic triangle" is a misnomer. The shape formed by the four figures on top is not a triangle, but a very slightly concave quadrilateral.

Note that the slope of the hypotenuse of the red triangle is 3 / 8 (0.375), but the slope of the hypotenuse of the teal triangle is 2 / 5 (0.400). They're close, but not exactly the same.

The sum of the areas in both figures is the same:
Red = 12 square units
Teal = 5 square units
Amber = 7 square units
Green = 8 square units
Total = 32 square units

The "hole," so to speak comes from the fact that due to the slightly differing slopes, where the top figure is concave, but the bottom is convex. The difference in areas of the dimple versus the bump is one square unit, the exact area of the so-called "hole."

Answer 4

The Impossible Triangle is an optical illusion.

This explanation is written in "plain English" since the geometric equasions behind this illusion are too complex to be understood by anyone but math majors. ;o)

Simply put: The triangle at the bottom is not really a triangle at all, it just looks like one. It's a fake triangle.

The dark green and the red triangle in the big fake triangle seem to form a straight diagonal line from the lower left corner to the upper right corner, like they do in the real triangle on top. But in reality it's not a straight line in the fake triangle.

Compare the real triangle on top with the fake triangle at the bottom. Take a close look at their upper diagonal sides and you will see that the way they cross the white grid in the background is slightly different.

There is an invisible dent where the dark green and the red triangle meet in the fake triangle. As strange as it may seem, the slight dent is enough to throw off the straight line, and enough to amount to the volume of the missing square.

xxx

Answer 5

It's just an optical illusion. If you draw it on a board and cut it and re-arrange it like the drawing you will notice that the diagonal side in the second position will not be a straight line and will be bulging out a little at the point where the two smaller triangles meet to compensate for the missing square so that the area remains 32.5 which is the area of the original triangle (5x13/2=32.5)

Answer 6

The extra square results from the assumption that the hypotenuses of the two smaller triangles are in an exact straight line. The discrepancy amounts to about 3% of the large triangle's area.
the area of the large triangle = (5*13) / 2 = 32.5 squared unit
so the discrepancy ammount = 1*100/32 % = 3.07%
if u notice
the first figure area is 32
the second one is 33
so there is a discrepancy of 1 square
and the mean value is 32.5
If u don't understand that well please Im
mabuhelwa

Answer 7

The figure isn't a triangle. Hence, the lower shape, if it were solid, has a larger area than the upper one. The Red triangle has a slope of 3/8 and the green triangle has a slope of 2/5 - close, but not the same. Area is conserved between the two shapes BECAUSE OF the gap :-)

Answer 8

It's because of how the triangle is shaped, by not being a real triangle.

The triangle you seem to see is of 32.5 ((5*13)/2).

They actually make an area of 32 units in the first part and 33 in the second if worked out from the blocks making them (area of each shape added together). This accounts for the missing block.

Answer 9

It's an optical illusion.

The hypotenues of both triangles seem to be straight lines but they're not.

The small triangle has an angle of inverse tangent of 2/5 or [21.8 degrees] as the angle, and the large triangle has an angle of inverse tangent of 3/8 of [20.6 degrees].

Therefore the hypotenues of both triangles are actually made of two separate lines. This accounts for the extra space that appeared to have come from nowhere.

Bottom line the outlines of both figures look similar but because of the different angles of the two smaller triangles, are not identical. That is where the extra square came from.

Answer 10

the slope on the 2 small triangles is not the same
the 3x8 one is 3/8 = .375
the 2x5 one is 2/5 = .4

so on the large triangles, the one with the hole has a bulged diagnal line

and the one without has a concave or dented line

if you drew a line across the full length it would not match the shape

if you add up the difference in each partial square along the diagnal line, the total would come out to exactly one square of area

I hope I am clear enough