Can anybody explain why on rearranging the triangle a hole appears?

Question

Hey, this is really puzzling me!!! check out http://web.mat.bham.ac.uk/N.M.Queen/history/hole.html

Answer 1

The length of the main hypotenuse is different for the two big triangles. One hypotenuse is longer than the other. This accounts for the one unit of "missing" area.

Look at the bottom triangle. Pay attention to where the green and orange sections meet. Their topmost intersection point is different when compared to the exact same spot in the big triangle at the top.

Answer 2

Essentially, the solution lies inherently in the incorrect assumption that you are actually looking at a triangle in the first place, as you aren’t. The gradient (slope of the line) of the green triangle is 2/5 = 0.4, however the gradient of the red triangle is 3/8 = 0.375. Therefore, you are looking at a polygon with four vertices as opposed to three. A triangle that has base 13 and height 5 has gradient 5/13, which is approximately equal to 0.384. By superimposing this triangle onto both of the polygons drawn in the question, you can clearly see where the additional square has come from in the first picture – its in white space in between the triangle and the object below – actually accounting for 0.5^2 additional area. Similarly, the second polygon has additional material above the diagonal, accounting for 0.5^2 additional area. Therefore, summing these two contributions together, we get 1^2 additional area in the second polygon – hence the need for a `hole’ in the material of size 1^2.

As a footnote, the lecturer in question who posed this problem takes great pleasure in the struggles of others to try to solve it, which seems ironic (and rather sad) given his apparent position as a lecturer. I also find it quite amusing that he has failed to pose the question correctly in the first instance, given that the objects aren’t actually triangles! It is a shame that people such as this represent the University of Birmingham - surely a quality university should seek to improve and develop students as opposed to ridicule?

Answer 3

Consider the gradient of the sloping line:
It is 5/13.
Now, 5/13 of 8 (the red triangle) is 3 + 1/13, not 3 as both diagrams seem to suggest! And the green triangle has a 1/13 deficit from the supposed height, 2 .

If you were to draw the second diagram accurately, the yellow and orange pieces would have edges just below the grid line; hence the hole. (Try calculating the area of the pieces.)

Answer 4

well, the red triangle is longer than the green triangle. and the brownish and yellow peices are cut such that the only match together. by placing the red triangle on top and the green triangle on the side, the rectangle made by the brown/yellow peices has to be longer. thus u separate the rectangle by moving the orange part left 2 and down 1. a hole is made because where the orange and yellow peices used to peice together, the orange only had 2 squares and the yellow had three. mainly its because the red triangle is longer than the green, so switching them makes everything else change. hope that helps :)

Answer 5

A very very good question!! I liked the problem a lot. Haven't got and answer, but gotta say the one Keith A said made most sense to me. nice going! one of the few problems i couldn't solve.!
The little trick is to realise the hypotenuse in the first trriangle is NOT i repeat NOT a STRAIGHT LINE. Keith A explains that well with trignometry! Wow, didn;t think that was an assumsion i was not sposed to make!

Answer 6

The only possible explanation I can come up with and I might be totally wrong is: (it says there are "hidden assumptions.") Well I was assuming that all the squares are the same size and maybe they aren't. Again, I may be totally wrong but maybe they are not entirely straight lines forming the grid.

That is the best I can do. Any one else with ideas?

Answer 7

The apex is pointed,and turned,produces a hole..so to speak. OK..we are talking pyramids here...that's the Pharaoh's burial chamber..stinks like hell in summer..

Answer 8

how can the hypotenuse be different if the lenght of the other 2 sides is the same and it is a right angled triangle????

BUT IT ISN'T A RIGHT ANGLED TRIANGLE
so the hypotenuses can be different

Answer 9

wow...this puzzles me too .....