Shouldn't Area be Preserved?

[peaceandlove]

What we know area always remain same but after seeing this picture , I doube Shouldn't Area been preserved?

First I though it was game of colour but I had even try this on papper , It work perfect.

Donot know where does the area gone? Strange Subhanallah

[glo]

I'm intrigued ...
Can anybody explain?

[aadil77]

wth is goin with that? height and width are same so....

[M.I.A.]

i was flummoxed so i went the google way lol,


i did not get the answer really, apparently the lines on the triangles are not straight and make up for the change in area.

i think if somebody made the puzzle in solid form and did a jigsaw that would be proof although the math says otherwise.

link to Dr.math

http://mathforum.org/library/drmath/view/61087.html

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[aadil77]

theres a whole wiki article on this - i still don't get it http://en.wikipedia.org/wiki/Missing_square_puzzle

[Tyrion]

After reading the wiki article and staring at the picture for a while, I can finally see the slight curve in each hypotenuse... The red and blue triangles aren't really triangles... Sneaky... Very sneaky indeed. :p

[peaceandlove]

I did not agree with people saying that "there are curves in lines" because I had tested it on graph paper my self.

Take a graph paper (with lot of boxes) and make the same drawing on it, then cut the colored pieces , and then rearrange them , you will still find space.

It something very strange and I think with no answer till yet, if you can answer you might win a noble prize for that.

[Tyrion]

I did not agree with people saying that "there are curves in lines" because I had tested it on graph paper my self.

Take a graph paper (with lot of boxes) and make the same drawing on it, then cut the colored pieces , and then rearrange them , you will still find space.

It something very strange and I think with no answer till yet, if you can answer you might win a noble prize for that.

The lines are slightly curved in the original picture. Compare the hypotenuse on each triangle with the other (using the boxes) and you'll see. You can't just recreate it on graph paper and expect it to work. :p

[Woodrow]

wHILE BOTH TRIANGLES APPEAR TO BE CONGRUENT THEY ARE NOT: IN FACT NEITHER IS AN ACTUAL TRIANGLE AS THE HYPOTENUSE IN EACH IS NOT A STRAIGHT LINE. EACH HYPOTENUSE IS MADE OF 2 LINES THAT DO NOT EXACTLY MATCH. NOTE WHERE I DREW ARROWS AND YOU WILL SEE THE OVERALL HEIGHT ALONG THE LOWER TRIANGLE IS HIGHER THAN IT IS ON THE UPPER ONE. THIS SLIGHT DIFFERENCE OVER ALL ADDS UP TO THE MISSING VOLUME.

WHILE THE INDIVIDUAL SHAPES THAT MAKE UP THE TRIANGLES ARE CONGRUENT WITH EACH OTHER THE TWO ARRANGEMENTS THAT LOOK LIKE EQUAL TRIANGLES ARE NOT CONGRUENT. NO AREA IS LOST. THE REARRANGING LOOKS SIMILAR BUT IT IS NOT IDENTICAL.

[GreyKode]

I think I found the answer..I mighr be wrong though.


The blue triangle hypotenuse has a slope = 2/5

The big red triangle hypotenuse has slope = 3/8

They are not equal and thus the composite triangle is actually not a triangle. The apparent hypotaneous of the composite triangle is not a straight line, thus u cannot apply the base*height rule on the composite triangle.

[GreyKode]

Oops didn't notice that woodrow gave the complete answer. Sorry brother woodrow :).

[glo]

wHILE BOTH TRIANGLES APPEAR TO BE CONGRUENT THEY ARE NOT: IN FACT NEITHER IS AN ACTUAL TRIANGLE AS THE HYPOTENUSE IN EACH IS NOT A STRAIGHT LINE. EACH HYPOTENUSE IS MADE OF 2 LINES THAT DO NOT EXACTLY MATCH. NOTE WHERE I DREW ARROWS AND YOU WILL SEE THE OVERALL HEIGHT ALONG THE LOWER TRIANGLE IS HIGHER THAN IT IS ON THE UPPER ONE. THIS SLIGHT DIFFERENCE OVER ALL ADDS UP TO THE MISSING VOLUME.

WHILE THE INDIVIDUAL SHAPES THAT MAKE UP THE TRIANGLES ARE CONGRUENT WITH EACH OTHER THE TWO ARRANGEMENTS THAT LOOK LIKE EQUAL TRIANGLES ARE NOT CONGRUENT. NO AREA IS LOST. THE REARRANGING LOOKS SIMILAR BUT IT IS NOT IDENTICAL.

So the answer is merely that one hypotenuse is slightly concave, the other slightly convex and the difference between the two makes up for the extra square?
And that the difference is so barely visible to the naked eye that one might not notice?

CLEVER!
Woodrow, Tyrion and Greycode, I liked your explanation by far better than the complicated mathematical one on the wiki page!

[peaceandlove]

Brother , GreyKode and Glo

The lines are all straight , the hypotnuse of both triangles are straight (not curve) , Just take a graph paper and try to draw a straight line trinagle as shown in figure and then re-arrange them , You will still find the space.

Once I also thought that the lines might not be strainght but those are straights.

As far as I know , Its not an optical Illusion as somebody might be thinking nor any tricks. It just a phenomenon which might not be described.

[GreyKode]

Indeed each hypotenuse is straight, the composite triangle's "apparent" hypotenuse however consists of 2 lines (the hypotenuse of the 2 triangles) that are not "colinear"(although this is not the most accurate term) i.e. they don't smooth out. The inequality of the two slopes is evidence for this .

The blue triangle hypotenuse has a slope = 2/5 > The big red triangle hypotenuse has slope = 3/8

[Woodrow]

Calculating the hypotenuse length of either large triangle we have a base of 13 and an altitude of 5 giving a hypotenuse of 13.928. The red triangle has a calculated hypotenuse of 8.544 and the blue one of 5.385 giving a total of 13.929 a difference of only .001 and too small to detect with any visual instruments. The difference can only be calculated, not visually discerned with readly available instruments. The calculated area for the large triangles is 32.5 basing the calculation on a base of 13 and an altitude of 5. The added area of the shapes give an area of 32. indicating a slight inward bend making up for the lost area in the top triangle. The bend is not perceivable with any visual instruments. You can only detect it mathematically.

The Bottom triangle has a combined area of 33 counting the addition of the blank square, indicating an outward bend that gained a 1/2 square in area.

In the small size of those triangles the bend is not visible, nor measurable except with the most sophisticated precision instruments. which few people would have access to. The bend is less than the thickness of the line drawing the hypotenuse. But the bend while not discernible to us humans is easily calculated and can be mathematically proven to be there.