Triangle Logic Puzzle

[MidNiteMysT]

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

ive been trying to figure this out but i dont get it. if you put it into photoshop and transfer the pieces, they are the same size. i looked at the explanations but they dont make sense. they make it seem like its made of liquid of if the angle is off, it makes a new square, but i dont see that at all. the way i think of it is, if a box is filled with cubes and it completely fills the box, no matter how you rearrange it, its gonna be full, theres no way to get around that, and thats why i dont understand how this picture works. is the last explanation right and i just dont understand it? or is there something more to it? im usually good with math problems and the like, but this has got me.

btw, leave your condesending know-it-all attitude at the door if you plan to have one. be polite and respectful please.

 

[PurdueRy]

Yes, The slope of the two lines composing the hypotenuse is not the same and therefore is not a triangle at all in the first place.

 

[dighn]

so old

the answer is that the triangles are really not the same size. compare the perimeters carefully

 

[lozina]

simple. take a piece ofpaper and line it up with the hypotenus (sp?) of both triangle. see the differencE?

 

[ijester]

Too really make it simple....

Count over five spaces from the right side on both. Look at the height of the hypotenuse. Second one is higher.

 

[tjaisv]

omg it's so eazy!

 

[OulOat]

Geometry > You

 

[troytime]

i must have smoked too much pot as a kid

when i take the image in photoshop, cut out the shapes in the top triangle
they lay perfectly over the matching shapes in the bottom triangle
angles and perimeters are really close to the same...not off far enough to leave one grid square empty

 

[troytime]

ok now i get it

bent hypotenuse

 

[PurdueRy]

Originally posted by: troytime
ok now i get it

bent hypotenuse

yay!

 

[OdiN]

Originally posted by: troytime
ok now i get it

bent hypotenuse

Yup. The grid is there ONLY to create an optical illusion.

 

[iamaelephant]

Are you guys high? The hypotenuse doesn't need to be bent - there is an equal distance on the bottom as there was before. Think about it. The red triangle is 8 squares across. The green shape is 5 squares across, for a total of 13 squares along the bottom. Now rearrange the yellow and green so they are face to face with the gap at the bottom. 5 squares across for green, plus two for yellow and 5 for the dark green triangle, and one gap = 13 squares. The triangle is identicle. You people think too much. Or am I missing something that's going to make me look like a total ass?

 

[MidNiteMysT]

but how could a little bend result in a whole missing square? its not like its made of liquid and you can make space to make a sqaure disappear. i do see that the hypo isnt a little bigger but i just dont see how a little bigger hypo can result in a perfect missing sqaure. if you photoshop it, you will see that the shapes fit pretty much exactly. im actually gonna try this on grid paper and see what happens.

if it was all one piece made of water or something, then yea, i could see how the longer hypo could result in a nice missing sqaure piece. but if its blocks like that, and you get some extra room, how do you make that into a missing piece? please explain further.

 

[JDrake]

yuck ebaums

 

[MidNiteMysT]

if you draw each shape out on grid paper to form each of the triangles, it works out. without bending the hypo.

edit: btw, yea i know this puzzle thing is old.

 

[Born2bwire]

Originally posted by: Falcon39
Are you guys high? The hypotenuse doesn't need to be bent - there is an equal distance on the bottom as there was before. Think about it. The red triangle is 8 squares across. The green shape is 5 squares across, for a total of 13 squares along the bottom. Now rearrange the yellow and green so they are face to face with the gap at the bottom. 5 squares across for green, plus two for yellow and 5 for the dark green triangle, and one gap = 13 squares. The triangle is identicle. You people think too much. Or am I missing something that's going to make me look like a total ass?

Quite simply, the first figure is not a triangle, it's a quadrilateral. The two shapes would be the same if an only if the red and green triangles were similar triangles (their corresponding angles are the same). However, the slope of the hypotenuse in the red triangle is lower than the slope of the green triangle. So the top triangle actually has a concave (curves in) hypotenuse. This means that it has slightly less area than if it was a real triangle. Now in the second figure, since the green triangle is on the bottom, the hypotenuse bends outward, or convex. So now you have more area than if it was a real triangle. The difference in area between the convex (more area) and concave (less area) hypotenuse figures is conveniently the same as an exact grid square. If you really need any more convincing, compute the area of each individual shape and look at them independently of each other and you'll realize that the sum of the areas of the individual shapes is not equal to the area that is computed if you assume the top figure is a triangle.

 

[Number1]

Quote

Quite simply, the first figure is not a triangle, it's a quadrilateral. The two shapes would be the same if an only if the red and green triangles were similar triangles (their corresponding angles are the same). However, the slope of the hypotenuse in the red triangle is lower than the slope of the green triangle. So the top triangle actually has a concave (curves in) hypotenuse. This means that it has slightly less area than if it was a real triangle. Now in the second figure, since the green triangle is on the bottom, the hypotenuse bends outward, or convex. So now you have more area than if it was a real triangle. The difference in area between the convex (more area) and concave (less area) hypotenuse figures is conveniently the same as an exact grid square. If you really need any more convincing, compute the area of each individual shape and look at them independently of each other and you'll realize that the sum of the areas of the individual shapes is not equal to the area that is computed if you assume the top figure is a triangle.[/quote]

Look like triangles to me.
I can't see any curves either.
Quadrilateral is an adjective, not a shape. Object has 4 sides.

Just count the number of squares and you will see that both triangles have the same dimentions.

 

[PurdueRy]

Originally posted by: Number1

Quote

Quite simply, the first figure is not a triangle, it's a quadrilateral. The two shapes would be the same if an only if the red and green triangles were similar triangles (their corresponding angles are the same). However, the slope of the hypotenuse in the red triangle is lower than the slope of the green triangle. So the top triangle actually has a concave (curves in) hypotenuse. This means that it has slightly less area than if it was a real triangle. Now in the second figure, since the green triangle is on the bottom, the hypotenuse bends outward, or convex. So now you have more area than if it was a real triangle. The difference in area between the convex (more area) and concave (less area) hypotenuse figures is conveniently the same as an exact grid square. If you really need any more convincing, compute the area of each individual shape and look at them independently of each other and you'll realize that the sum of the areas of the individual shapes is not equal to the area that is computed if you assume the top figure is a triangle.

Look like triangles to me.
I can't see any curves either.
Quadrilateral is an adjective, not a shape. Object has 4 sides.

Just count the number of squares and you will see that both triangles have the same dimentions.[/quote]


quad·ri·lat·er·al Pronunciation (kwdr-ltr-l)
n. Mathematics
A polygon having four sides.

It is a shape and the shape in the problem has 4 sides

 

[JujuFish]

Originally posted by: PurdueRy

Originally posted by: Number1
Look like triangles to me.
I can't see any curves either.
Quadrilateral is an adjective, not a shape. Object has 4 sides.

Just count the number of squares and you will see that both triangles have the same dimentions.

quad·ri·lat·er·al Pronunciation (kwdr-ltr-l)
n. Mathematics
A polygon having four sides.

It is a shape and the shape in the problem has 4 sides

:laugh:

 

[Number1]

Originally posted by: PurdueRy
quad·ri·lat·er·al Pronunciation (kwdr-ltr-l)
n. Mathematics
A polygon having four sides.

It is a shape and the shape in the problem has 4 sides

I stand corrected, now where is the quadrilateral?
Still look like triangles to me.

 

[JujuFish]

Originally posted by: Number1

Originally posted by: PurdueRy
quad·ri·lat·er·al Pronunciation (kwdr-ltr-l)
n. Mathematics
A polygon having four sides.

It is a shape and the shape in the problem has 4 sides

I stand corrected, now where is the quadrilateral?
Still look like triangles to me.

Three of the corners are the vertices of what appear to be a triangle, the fourth is where the green and red triangles touch.

 

[Number1]

Originally posted by: JujuFish


Three of the corners are the vertices of what appear to be a triangle, the fourth is where the green and red triangles touch.

???????????????????????????
Still look like triangles to me. Triangles that had their components rearanged to fit into the same shape.

 

[Born2bwire]

Originally posted by: Number1

Originally posted by: JujuFish


Three of the corners are the vertices of what appear to be a triangle, the fourth is where the green and red triangles touch.

???????????????????????????
Still look like triangles to me. Triangles that had their components rearanged to fit into the same shape.

The reason why it is not visually apparent is because they cheated by using thick lines for the triangles. The actual deviation from the true hypotenuse is covered up.

Text

I have drawn in using lines of a single pixel width to demonstrate what I am talking about. The red line represents the true hypotenuse of a 13x5 ratio right triangle. The four other black lines that make up the other "hypotenuses" are the "hypotenuses" that are created by the two figures in the picture. You can clearly see how one "hypotenuse" distends from the true hypotenuse and the other is bent below it. Now if you calculated the area between the four black lines you should get the area equal to one of the grid squares. This accounts for your mysterious hole that pops up.

If you drew the true hypotenuse on the figures in the original pic, it would be almost completely hidden by the thick lines of the original triangles, which is why it is difficult to gauge the slight change in slope (from .375 to .4 in the top polygon) visually. It is only until you look at the ratio of the sides of the red and green triangles that you realize that the triangles are dissimilar. Hence, the top polygon is not a triangle but a quadrilateral.

 

[DrPizza]

Hey, thanks for that image born2bwire...

Smart: figure it out on their own.
Average: figure it out when someone tells them it doesn't form a triangle.

What's the category when they still think it's a triangle after being told it's not??


When people sign up on these forums, in addition to the box you check that says you agree to the TOS, there should be a couple of other questions... "Where'd the square go?" and "Will the plane take off?"

 

[Born2bwire]

Originally posted by: DrPizza
Hey, thanks for that image born2bwire...

Smart: figure it out on their own.
Average: figure it out when someone tells them it doesn't form a triangle.

What's the category when they still think it's a triangle after being told it's not??


When people sign up on these forums, in addition to the box you check that says you agree to the TOS, there should be a couple of other questions... "Where'd the square go?" and "Will the plane take off?"

Sweet God I cannot wrap my head around how that plane on conveyor belt thread just keeps going on and on. It needs to die a swift death. I do remember that your post was pretty much all that needed to be said.