Could anyone give me a logical and real answer to this geometrical problem?

Look at the two triangles. They look similar, but if you compare the ratios of the legs you will see that they are not actually similar. The ratio for the red triangle is 3:8 and the ratio for the blue-green triangle is 2:5. The basic trick is that the hypotenuse is not really a straight line even though it appears so because it is very close to being straight.

If you look closely at the hypotenuse of the triangle, you still wouldn't see it. However, the two 'triangles' are not triangles at all. The hypotenuse is not a straight line. To prove this, graph the line. The 'triangle' has a height of 5 and a length of 13. A line with a slope of 5/13 would not cross another destinguishable point on the graph between the two end points of the 'triangle' simply because 5/13 cannot be simplified. If you look closely at the first shape you can see a definite point on the hypotenuse 2 grids down and 5 to the left of the top. If you also look on the second shape you can see a point 5 grids right and 2 grids up from the bottom. This small, almost indistinguishable difference hides the last "missing square" in the area that was lost when changing the line on the first shape to the second. All but a very ingenious trick.

YOu are Wrong.

There you do not have 2 triangles of the same exact size.

In fact there are no triangles at all, and they are both 4-sided figures.

Why?

The sides of the red piece are 3 and 8.

The sides of the green piece are 2 and 5.

The top-right angle of the red one is arctan(8/3).

The top-right angle of the green one is arctan(5/2).

since arctan(8/3) is not equal to arctan(5/2), the 2 angles are not the same.

Actually, 8/3 > 5/2 and arctan 8/3 > arctan 5/2.

THEREFORE, The topright angle of the red piece is larger than

that of the green one.

In fact, the difference of the angles is 1.245 degrees (cannot be clearly seen by the naked eye).

On the first figure, since you know that the topright red angle is greater than the topright green angle, it is convex.

On the bottom figure, since you know that the topright green angle is less than the topright red angle, it is concave.

If you will look at both "hypotenuses" of the figures, you will see that they are not equal.

The top figure is a pit. The bottom one is a hill.

An extra space can fit in the difference of the areas of the hill and the pit.

^_^

Neither whole figure is a triangle. The hypotenuses aren't a straight line. Look at the smaller blue triangle. It's short leg is 2 squares long and it's long leg is 5 squares long. Now look at the larger red triangle. The short leg is 3 squares long or 50% longer than the short leg of the blue triangle. In order for it's hypotenuse to be at the same angle as the smaller triangle it's longer leg would have to be 50% longer than the long leg of the blue triangle or 7 1/2 squares long, not 8 squares as it is. therefore you are really comparing the area of two quadrangles that both look awful darn close to a triangle, but not quite.

Is very simple. The Rectangle (two blocks) in the 1st triangle is 3x5 = 15 blocks. The Rectangle in the 2nd triangle is 2x8 = 16 blocks. Other shapes the same. Triangle angles also slightly different. overall 13x5 at gradient, little triangle at 2x5, big at 3x8

Therefore, angle is slightly different for the triangles overall.

If you look closely you can see it, that the 1st triangle dents inward, and the 2nd dents outward, leaving space for the single block.

a triangle is made of three line segments formed in a certain way, not of a toy block or other solid. also, the triangles are of the same size but possibly not the same shape. think about it.

in the second triangle (with the hole) the hypoteneuse is not quite straight. Very slight, but it makes a difference. Probably other slight adjustments too.

those are not triangles. the long side caves in or buldges out, generating extra 1 square of area.

the top one has a steeper angle. moving the triangular pieces changes the angle to be less steep and somehow that creates that space.

Because the triangles are affixed to your ears.