http://www.ebaumsworld.com/trigrid.html

Thanks!

The diagonal "line" is not really a line. check the small angle of the small triangles.

What do you mean? It sure looks like a 'line' to me. And how do I measure the angles? They look very similar.

OK, Chout,

There are two triangles to compare. The green one (with side A=2 and side B=5) and red one (with side A=3 and side B=8). Now a basic law of trigonometry says that when you take Side A and Divide it by Side B you will get a number known as a tangent. So for the green one, the tangent is 0.40 and for the red one, 0.38. These numbers are called the tangent of the angle opposite the side A of each triangle.

If these angles were identical, their tangents would be identical, therefore that line is not straight (Another geometrical fact). If you drew this on graph paper and put a straight edge along side the hypotenuse, it would be obvious.

Now where did that hole come from? It actually comes from that area outside of the triangle along the hypotenuse. You can calculate it and it will show to be 1 square unit, if you compare it to the initial conditions of the demonstration. It's based on an old Sam Lloyd Puzzle from the turn of the last century...one of the greatest puzzle masters in history.

Steven

As Steven pointed out, the two triangles are not similar, and so what looks like one diagonal line is in fact a bit bent.

What makes this work so well is the thickness of the black lines used to draw the figure.

Were you to carefully take a ruler and cut out the shapes from a sheet of paper, color the shapes and remount on a sheet of graph paper you could see where the evasive square goes.

The map is not the terrain indeed! sometimes the lines we draw to make a map hide critical information.

Chout,

Here is some additional information. The top triangle we'll call No. 1. The area of a triangle is equal to 1/2 times the base times the height; so the red triangle is 12 square units and the green is 5 square units. Added to the 15 square units of the rectangles the total area is 32.00 square units.

This means that the area measured is 0.5 units SMALLER than the area of a 5 by 13 triangle (32.5 square units)

Now, and this will take some thinking...take Triangle No. 2 and rotate it 180 degrees or flip it upside down, in the vernacular. Butt the triangles along their hypotenuses together to form a rectangle and they will fit perfectly together, no spaces. What this means is that area along the hypotenuse of triangle No. 1, which was missing is now, filled by the hypotenuse of Triangle No. 2...meaning that Triangle No. 2 is 0.5 units LARGER than a 5 by 13 triangle. So triangle 2 is filling in the missing area of triangle 1 and, in addition, adding another 0.5 units along the hypotenuse. The unaccounted for area (now 1.0 units) is shown in the rectangular area.

I know this is confusing, but hopefully you can follow my convulted...yet somehow, logical thinking.

Steven

Ya now I understand.

The hole is not whole, so it's not really a whole hole. It's only a part hole because the diagonal "line" is not really a line. And therefore, triangle 2 is filling in the missing area of triangle 1 and, in addition, adding another 0.5 units along the hypotenuse. Simple

Man, ain't math great.

I always thought it was a hole in the space-time continuum. Or the porthole to an alternate dimension. Or the passageway to that mysterious place where one of my socks always seems to go when I put them in the dryer..
:cyclops:

I just watched stargate, (stop laughing) I wonder if your dryer is like a startgate and that's where your socks are going..?

The top line of the triangle is slightly curved to the point where it frees up the space of another square.

Oh guys this is so simple, at least for me I immeadetly looked all those parts separetely and everything was so very clear at that point

I see whats going on here ... Ill see if I can explain it. you can arrange the 2 triangles to leave 2 different "holes" to be filled by the other 2 objects. As a shape gets closer to to a circle the less perimeter you need for the same area (eg if you were building a kennel and wanted to maximize the area with a set amount of fencing you would build it in a circle ... or if you were limited to straight edges you would build a perfect square). So one of the holes is closer to being square (thus closer to being a circle) leaving more area to be covered), but we have a fixed amount of area to fill the hole with .. thus the leftover uncovered part of the hole. Im sure you could generate a proof using the calculus if you had the time and inclination ...

neat effect though!

No need for trickery like bent or curved lines here, btw

unless I am missing something ... looks clear to me though ...

Yikes ... I was way way wrong ... shoulda listened to you guys

looks like they did pinch a half a small square with bent lines ...

no more of these at night for me ...