A geometry problem for the mathletes

[pokerfan]

[XemaSab]

1. The dark green and red triangles are not similar

We're being tricked into thinking they are, and that the top and bottom larger triangles have the same area. Which they clearly don't.

[DS1]

2. They are 8x3 and 5x2 units in both

[LynzM]

4. Right, but not that way

The ratio of the sides between those two is not the same, hence the upper 'triangle' is slightly convex, and the bottom slightly concave (or reverse, cannot remember since I cannot see the pic now.)

Maybe?

[pokerfan]

6. You can't see the pic?

[DS1]

7. nope

just sliced it in two in pshop just to make sure

perfect overlay on all objects

[XemaSab]

14. Cave = innie

;)

[XemaSab]

5. Calculate the slope of the line for the green and red triangles

They have different slopes, and they're not the same.

[redqueen]

8. You can just look at them and see the slope isn't the same.

But it's hot that you calculated it. :P

[DS1]

9. see

[XemaSab]

12. Let me break it down for you:

The slope of the hypotenuse is directly related to the angles on each corner of the triangle. Similar triangles have the same angles at each corner.

The green triangle is 5 long and 2 high.

The red triangle is 8 long and 3 high.

Slope is rise over run.

2/5= .4

3/8 = .375

If the triangles were similar, their hypotenuses would have the same slope. The slope is not the same, so they're not similar and we are being fooled by our eyes.

[DS1]

13. I'm not arguing that

I'm just pointing out that the triangles are the same in each pic. Perhaps we crossed messages somewhere

[XemaSab]

15. No worries

:hi:

[Systematic Chaos]

16. First of all they are NOT triangles.

It's an optical illusion.

They first pic actually shows a quadrilateral with an unmarked point where the two triangles meet. This is because, as already explained, the slopes of those two line segments are NOT THE SAME. Your eyes might tell you they are, but they arent, so AB is not actually a line, so ABC, by definition, is NOT a triangle.

It would be like taking a perfect square of x units per side, cutting it into several irregular shapes, and then wondering why you don't get a square when you rearrange those random shapes in a different fashion.

The sum of the areas of the shapes in both diagrams is the same, yes, but the assumption that ABC is a triangle is false; in fact, it is impossible to form an actual triangle from the pieces which constitute ABC.

Now do you understand?

[Rabrrrrrr]

19. Yes, the red are the same and the blue are the same; but the red and blue are not similar

similar in the mathematical sense of having the same interior angles.

They look like it - and that's the crux of this particular optical illusion; they are drawn very closely to similarity, but are just enough different to end up with that missing 1 square in area when re-arranged.

That missing box is made up by extra area in the difference between the top entire fake triangle's 'hypotenuse' and the bottom entire fake triangle's 'hypotenuse'.

[flvegan]

3. The hole? Stray bullet from the latest GD gun thread.

[Systematic Chaos]

10. AB in the first diagram is not a true line.

Neither is DE in the second diagram.

The slope of the red triangle's hypotenuse is 8/3 or 2.667:1 while the green triangle's hypotenuse is 5:2 or 2.5:1
That extra square comes about - somehow - due to that difference in slopes and the way the pieces fit.

[Inchworm]

11. Proof

That there is more than one way to skin a cat.

:hide:

[kentauros]

17. The hole has nothing to do with the triangles.

Look at the grid, not the slopes. None of the shapes changed in any way when they were rearranged. BUT, the area that the two rectangular pieces occupied did change. The rectangular pieces also dropped down one square in order to handle the longer side of the red triangle and the short side of the teal triangle. Notice, too, that the area these two shapes cover have a common point on the hypotenuse. This is key to the area change. The difference between the two points also is key to why the hole shows up, and has to do with the difference in the area between the two points. That is, the area not used in the rectangular area change. It's anotehr triangle, so look at it and compare its area to that of the hole.

In the first triangle, the rectangular shapes occupy an area of 3x5 squares.
In the second triangle, the rectangular shapes occupy an area of 2x8 squares.

15 squares versus 16 squares, thus the hole :D


I hope I've explained that well enough...

[UndertheOcean]

18. AB is not a straight line , you can't just eyeball geometry