That damn triangle
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The hypotenuses of the triangles are not at the same angle. The hypotenuse of the large triangle is arctan(3/8) = 20.6 degrees to the baseline, while the hypotenuse of the small one is arctan(2/5) = 21.8 degrees. The top shape is therefore not a triangle, so there's no reason why they should fit together perfectly in the bottom shape. Try it yourself :) Ryan Being little and getting pushed around by big guys all my life I guess I compensate by pushing electrons and holes around. What a bully I am, but I do enjoy making subatomic particles hop at my bidding - Roger Wright (2nd April 2003, The Lounge)
Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late - John Nichol "Point Of Impact" -
I've reached the same conclusion: the first figure is not a triangle, but a quadrilateral.
Shake, Courage. Shake.
KaЯl wrote: the first figure is not a triangle, but a quadrilateral Exactly :) Ryan Being little and getting pushed around by big guys all my life I guess I compensate by pushing electrons and holes around. What a bully I am, but I do enjoy making subatomic particles hop at my bidding - Roger Wright (2nd April 2003, The Lounge)
Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late - John Nichol "Point Of Impact" -
I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
John Honan wrote: And both triangles ARE the same, AND the hypotenuse is a straight line. No, the two main "shapes" are not triangles and their "hypotenuse" is not a straight line. Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. (ie. They must have exactly equal angles). Red triangle angles: 90, x and y where tan x = 3/8 and y = 90 - x x = 20.556 degrees y = 69.444 degrees Dark green triangle angles: 90, x and y where tan x = 2/5 and y = 90 - x x = 21.801 degrees y = 68.199 degrees Let's take it a step further: Main "shape" angles: 90, x and y where tan x = 5/13 and y = 90 -x x = 21.037 y = 68.963 This "puzzle" is purely an optical illusion.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching. -
The hypotenuses of the triangles are not at the same angle. The hypotenuse of the large triangle is arctan(3/8) = 20.6 degrees to the baseline, while the hypotenuse of the small one is arctan(2/5) = 21.8 degrees. The top shape is therefore not a triangle, so there's no reason why they should fit together perfectly in the bottom shape. Try it yourself :) Ryan Being little and getting pushed around by big guys all my life I guess I compensate by pushing electrons and holes around. What a bully I am, but I do enjoy making subatomic particles hop at my bidding - Roger Wright (2nd April 2003, The Lounge)
Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late - John Nichol "Point Of Impact" -
I agree with you that it's not as easy as it looks. Here's my attempt to explain. Good luck. The base of the red triangle is 8, and the base of the dark green is 5. Together, in either order, they add to a base of 13, so the results appear of equal size. If you remove the red triangle from the top image, you are left with a dark green triangle that is the same size horizontally as both the brown and light green triangles, that being 5. In the bottom image, the brown and light green triangles don't line up evenly. They can either be 5, as in the top image, or 8 with a hole, as in the bottom image. The problem comes from the overlap between these two triangles, being 2 in the top image and 2+1 in the bottom. The 3 square of brown can't overlap the 2 squares of light green without adding 1, which is the hole. In the top image, the overlaps match and the base is 5 to match the dark green. In the bottom image, since the overlap isn't even, it appears with a hole but makes a base of 8 to match the red. Ouch. Dave "You can say that again." -- Dept. of Redundancy Dept.
Thanks for trying to explain. But I still can't see how surface area just disappears between the two triangles, as they're both 32.5 This is how far I've taken it: http://www.silveronion.com/images/squares2.gif[^] John www.silveronion.com[^]
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
The global shapes both look like triangles but actually aren't triangles. Also, if you could put them one on the other you would see that they do not match because of their "hypothenuses" that aren't straight lines. They do not have the same area. Enjoy yourself!
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John Honan wrote: And both triangles ARE the same, AND the hypotenuse is a straight line. No, the two main "shapes" are not triangles and their "hypotenuse" is not a straight line. Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. (ie. They must have exactly equal angles). Red triangle angles: 90, x and y where tan x = 3/8 and y = 90 - x x = 20.556 degrees y = 69.444 degrees Dark green triangle angles: 90, x and y where tan x = 2/5 and y = 90 - x x = 21.801 degrees y = 68.199 degrees Let's take it a step further: Main "shape" angles: 90, x and y where tan x = 5/13 and y = 90 -x x = 21.037 y = 68.963 This "puzzle" is purely an optical illusion.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching.Okay thanks Mike (and all the other people who replied) - It's starting to sink in, slowly... The whole illusion is based around the small differences in these angles then, and the relatively large affect they have on surface area John www.silveronion.com[^]
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
I don't understand your confusion. A rectangle thats 5 units by 3 units is 15 units. A rectangle thats 8 units by 2 units is 16 units. That's sort of really basic geometry isn't it? The triangles have nothing to do with it, ignore them.
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Okay thanks Mike (and all the other people who replied) - It's starting to sink in, slowly... The whole illusion is based around the small differences in these angles then, and the relatively large affect they have on surface area John www.silveronion.com[^]
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I don't understand your confusion. A rectangle thats 5 units by 3 units is 15 units. A rectangle thats 8 units by 2 units is 16 units. That's sort of really basic geometry isn't it? The triangles have nothing to do with it, ignore them.
Stan Shannon wrote: A rectangle thats 5 units by 3 units is 15 units. A rectangle thats 8 units by 2 units is 16 units. My confusion is because they should actually both be 15.5 units in area, not 15 and 16 as the illusion would have you believe. John www.silveronion.com[^]
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Thanks for trying to explain. But I still can't see how surface area just disappears between the two triangles, as they're both 32.5 This is how far I've taken it: http://www.silveronion.com/images/squares2.gif[^] John www.silveronion.com[^]
John Honan wrote: I still can't see how surface area just disappears It's not a question of surface area. It's not a matter or size, or shape, or angles, or hypotenuses, or any of that. It's a matter of 2 shapes (brown and light green) being made to line up with one size triangle (dark green) and then being made to line up with a differently sized triangle (red) ... and then expecting the results to match up the same way. Think of it this way. If you have 5 pounds of stuff in a 5 pound bag, and then put that 5 pounds into a bigger bag, isn't there going to be some empty space? Don't let the mathematics get in the way of seeing the simple solution. In your thinking, replace the red triangle with the value "8" and replace the dark green triangle with the value "5". Now, position the light green and brown triangles so they have a base of 5 (top image). Then position the triangles so they have a base of 8 (bottom image). How can you orient them any differently and still expect them to agree? You can't. There's no trick to it. At this point, it's not an optical illusion. You just can't get 5 pounds of stuff into an 8 pound bag without having some empty space. When you compare 2 results to 2 different answers and expect them to be the same, you aren't seeing the problem, and it doesn't have anything to do with surface area, size, angles, etc. And yes, it was evil that they brought this up again. Dave "You can say that again." -- Dept. of Redundancy Dept.
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Stan Shannon wrote: A rectangle thats 5 units by 3 units is 15 units. A rectangle thats 8 units by 2 units is 16 units. My confusion is because they should actually both be 15.5 units in area, not 15 and 16 as the illusion would have you believe. John www.silveronion.com[^]
You are allowing yourself to be confused by superflous details. The base of the one rectangle is 8 and its height is 3, the base of the other rectangle is 5 and its hight is 2. Obviously, you are going to get a different sized rectangular area depending upon how you associated those two triangles. 5 * 3 = 15 and 8 * 2 = 16. Thats all the math you need.
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
Count the number of blocks and you will see where the hole comes from. The number of blocks on the bottom row for whole triangle is 13. If you look at the bottom rows for the green triangle, yellow block and green block and sum them then the total is only 12. :cool:
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
the angle on the greenblue block is 5/2 and the angle on the red block is 8/3 you can see the broken hpyo hmmm hypothumhmmh (you know what i mean) here: http://www.compona.com/fake.gif[^]
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
Here is the answer. angle 1 and 2 are NOT the same!!! Check the picture with clues. See the blue line I drew? See how on one there is a gap and on the other it cuts through? http://www.chepel.com/picture.jpg By number the answer is - 5:2 it not the same proportion as 8:3 for the red and green triangles.
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John Honan wrote: And both triangles ARE the same, AND the hypotenuse is a straight line. No, the two main "shapes" are not triangles and their "hypotenuse" is not a straight line. Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. (ie. They must have exactly equal angles). Red triangle angles: 90, x and y where tan x = 3/8 and y = 90 - x x = 20.556 degrees y = 69.444 degrees Dark green triangle angles: 90, x and y where tan x = 2/5 and y = 90 - x x = 21.801 degrees y = 68.199 degrees Let's take it a step further: Main "shape" angles: 90, x and y where tan x = 5/13 and y = 90 -x x = 21.037 y = 68.963 This "puzzle" is purely an optical illusion.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching.Mike Mullikin wrote: Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. I think this is true if you are comparing the red triangle to the dark green triangle. But, in this problem, we don't need to compare the 2 triangles to each other, we need to look at them together. Together, they are 8+5 horizontally and 3+2 vertically. In another arrangement, they are 5+8 horizontally and 2+3 vertically. I don't see that it makes any difference, because together, the result is the same. Dave "You can say that again." -- Dept. of Redundancy Dept.
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the angle on the greenblue block is 5/2 and the angle on the red block is 8/3 you can see the broken hpyo hmmm hypothumhmmh (you know what i mean) here: http://www.compona.com/fake.gif[^]
you beat me to it :mad:
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The hypotenuses of the triangles are not at the same angle. The hypotenuse of the large triangle is arctan(3/8) = 20.6 degrees to the baseline, while the hypotenuse of the small one is arctan(2/5) = 21.8 degrees. The top shape is therefore not a triangle, so there's no reason why they should fit together perfectly in the bottom shape. Try it yourself :) Ryan Being little and getting pushed around by big guys all my life I guess I compensate by pushing electrons and holes around. What a bully I am, but I do enjoy making subatomic particles hop at my bidding - Roger Wright (2nd April 2003, The Lounge)
Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late - John Nichol "Point Of Impact"I just got it also, the slopes are not the same, damn maths , I hate it, so simple yet, so magical! ;) I found that site the other day, the Eric Weisstein's World Of Mathematics (on wolfram site ) http://mathworld.wolfram.com/[^] and http://mathworld.wolfram.com/TriangleDissectionParadox.html[^] It's so great that I might one day start having fun with maths,
Maximilien Lincourt For success one must aquire one's self
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Mike Mullikin wrote: Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. I think this is true if you are comparing the red triangle to the dark green triangle. But, in this problem, we don't need to compare the 2 triangles to each other, we need to look at them together. Together, they are 8+5 horizontally and 3+2 vertically. In another arrangement, they are 5+8 horizontally and 2+3 vertically. I don't see that it makes any difference, because together, the result is the same. Dave "You can say that again." -- Dept. of Redundancy Dept.
David Chamberlain wrote: I think this is true if you are comparing the red triangle to the dark green triangle. But, in this problem, we don't need to compare the 2 triangles to each other, we need to look at them together. No, they MUST be compared. In each "shape" the two triangles are oriented similarly to the "shape". In order for the "shape" to be considered a triangle the two triangles MUST have EXACTLY the same angles. Not approximately the same angles, they MUST be PERFECTLY, EXACTLY the same! They are not, so the two shapes are not triangles and there should be no expectation of fitting the pieces together as if they were. I simply can't make it any clearer. :~
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching. -
the angle on the greenblue block is 5/2 and the angle on the red block is 8/3 you can see the broken hpyo hmmm hypothumhmmh (you know what i mean) here: http://www.compona.com/fake.gif[^]
Thanks :cool: John
