Regarding triangles: I figured it out! I first thought I had the answer and I started to explain it when during the explanation I realized that it didn't answer the question at all. Then I was confuzeled.
So I took the logic approach. Every small part of the overall triangle is exactly the same dimensions in both triagles. That would mean the sum of the areas of the smaller pieces should match the area of the overall triangle. Right? Ah, but that would only be true if both large overall triangles are indeed really triangles and are exactly the same.
The overall shapes certainly look the same to the eye, but they are not, and in fact they are not really triangles. In order for either of the overall shapes to be real right triangles the two internal triangles would have to have the exact same proportions (same angles). They don't. The smaller triangle has the angles roughly 38,52, and 90 degrees. The larger internal triangle has the angles roughly 35, 55, and 90 degrees (atan(2/5) != atan(3/8)). As you can see they are not the same angles. Close, close enough to trick the eye, but no cigar, not the same. Because they don't match also would mean the hypotenuse of the outer shape can not be a straight line, and therefore not really a hypotenuse at all, and also means the outer shape can not be a triangle. Hold a straight edge up to the hypotenuse of the overall shape and see that it is not exactly straight.
The top "shape" actually has a smaller overall area than the bottom shape. The sum areas of all of the pieces are equal in both large shapes and equal to the overall area of the top shape but the sum is smaller than the area of the bottom shape and thus accounting for the space in the bottom shape. Pretty wacky!
To mathematically figure out the answer you need to actually make real triangles out of the shapes. You can do this by drawing a line from the point on the "hypotenuse" where the two small triangles touch down to the right angled corner. You should now have a total of 4 right triangles that take up the full area of each large shape. With these 4 triangles you can easily figure the area of the total shape. Then compare those areas with the areas of the 4 smaller shapes:
Internal Shapes:
Green Triangle = 5U^2
Red Triangle = 12U^2
Yellow = 7U^2
Green = 8U^2
Total: 32 Square Units
Upper Large Shape divided into 4 right triangles:
Green Triangle = 5U^2
Red Triangle = 12U^2
New Triangle 1 = 7.5U^2
New Triangle 2 = 7.5U^2
Total: 32 Square Units
Lower Large Shape divided into 4 right triangles
Green Triangle = 5U^2
Red Triangle = 12U^2
New Triangle 1 = 8U^2
New Triangle 2 = 8U^2
Total: 33 Square Units
As you can see the bottom shape is actually one unit larger in area than the top shape thus creating the empty space of one unit. Had to dig way back in the memory banks for this one, been many a years for this old dog. But at least I remembered "base * height / 2" for figuring the area of a right triangle.
Also with the repositioning of the two triangles between the two shapes you go from having a rectangle space that is 3x5 (15U^2) to a rectangle of 2x8 (16U^2). Got any more like this? This one was fun. I can't believe I put this much into it. I hope I wasn't the only one that was stumped for a few minutes...
[ January 19, 2003: Message edited by: void main ]
