Is it possible for a right angled triangle with sides 3 and 4 units long to have a hypotenuse 6 units in length?

I’m not quite getting you. It isn’t actually a triangle when the hypotenuse has these indentations, right? The hypotenuse isn’t a straight line as you describe it. If the other sides are 3 and 4, the hypotenuse isn’t straight until it equals 5. So at 7, 6, and 5.00001, you don’t have a straight line.

Unless I’m misunderstanding what you’re suggesting.

It’s difficult to explain without a drawing but you have two things converging, the hypotenuse of the triangle and the stepped triangles, one 5 units in length and the other 7. With the indentations you do not have a line or a triangle that is true but as the steps become more numerous it tends towards a straight line and a triangle and the length of 7 units will become 5.

On thinking it over however it seems the fractal triangles will remain just that unto infinity and will never ‘converge’ on a straight line though whether this is really an answer I don’t know.

Maybe it could be stated that the limit as x becomes a hypotenuse equals 5.

Yes but it isn’t gradually tending towards 5, it remains at 7 however finely you draw the triangles. It would have to jump from 7 to 5 which doesn’t seem possible therefore the series of triangles will never form a straight line. Where is a mathematician when you need one!

It doesn’t change gradually? Wow. So as each indent gets smaller, more triangles are added to keep the length 7? Now I see what you’re saying. It is 7, then if it became a “pure” line it would jump through 6 to 5. I have no mathematical knowledge to explain that.

This is fractal geometry?

Let me state this: if the hypotenuse is 7 and comprised of increasingly smaller triangles, the other sides are likely not 3 and 4 if similarly comprised. So this hypothetical triangle doesn’t quite fit the classic Pythagorean theorem.

@cockswain You’re right, it would no longer be a triangle.

Undefined units are silly things. I don’t suggest you work with them if you can ever avoid it.

But to answer the title question, no.

The number of units in your hypotenuse is determined not by its length, but by the size of the indents. Smaller indents mean more indents, regardless of length.

A right triangle with sides 3 and 4 would have hypotenuse 5. To change the length of the hypotenuse you would need to change the length of the other sides, while keeping the angles constant.

The only way to get a right triangle with hypotenuse 6 is to use non-whole numbers. 3 by 5.1something or 4 by 4.4something would do it. But if you’re not shrinking the triangle on a scale with units small enough to hit the right proportion (you will need accuracy to four or five places beyond 0) you will skip entirely over 6 as a possible hypotenuse length.

If you want to talk about a triangle made out of blocks, so the hypotenuse is a number of right angles like a stair-case, then provide information about the size of the boxes, or at least how they scale to the triangle.

Let me know if this makes any sense, and if it doesn’t I will try to clarify.

@notdan That’s right a staircase describes it well but I was imagining the individual steps becoming smaller and smaller and tending towards a smooth line when the number of steps is infinite.

If you’re going to define the structure of the side as a series of steps, it will always be a series of steps, and never smooth. You’re cramming more and more steps into a given space, they will get smaller but the steps will never disappear. (To the naked eye, perhaps, but never to math!)

An infinite number of steps is as incomprehensible as any other infinite value.

Hi. Are you tallking about something like this:
http://img695.imageshack.us/img695/5514/phospho.png
in the current setup, the total of the vertical sides of the small triangles is 4, like the height of the triangle, and the total of the horizontal sides is 3, like the base of the large one. The hypothenuse is 5 for both. While individual values get smaller as you increase the number of triangles, the totals for height, base and hypothenuse still remain as 3,4 and 5, just that at some point it will seem like the two lines converge

ahhhh now i see what you are suggesting.
for the small triangles, total base + height = 7
and total hypothenuse for all small triangles = 5
so, as you keep making the sides smaller, at one point, the difference between the sides will be negligible, so each side will be so small that it will be impossible to tell the difference between the length of the hypotenuse and the other sides. but that would mean that there would be no seeable difference between what should be different
so in that case, how can 5 [length of hypotenuse] = 7 [length of total sides]
the only conclusion i can come up with is that, at that point, number of triangles will tend to infinity, and so, even though each individual side will be tiny, and almost the same as each other, when you multiply even something so tiny by something close to infinity [the number of triangles], the total length will still come to 5 for hypotenuse, and 7 for the sides

The more I think about it, I like the statement (with regards to this specific example): The limit of the length of the hypotenuse as x approaches infinity is 5. X being the number of indents.

@bigjay I was imagining the sides staying the same but the number of triangles (as in your diagram) increasing while getting smaller. The total length and heights of all the tiny triangles added together will remain 7 even when the triangles are so small they cannot be seen and it looks like a straight line hypotenuse.

I think we have worked out that the hypotenuse made of tiny triangles is not a hypotenuse at all but a fractal masquerading as a hypotenuse and has a length of 7 units.

Well we didn’t need to call out the mathematicians after all. Reminds me of the time I cleared out a blocked drain by myself and didn’t require a plumber.

I think we have worked out that the hypotenuse made of tiny triangles is not a hypotenuse at all but a fractal masquerading as a hypotenuse and has a length of 7 units.

Actually I like that statement better than the one I put above yours. I’m going with that.

“I think we have worked out that the hypotenuse made of tiny triangles is not a hypotenuse at all but a fractal masquerading as a hypotenuse and has a length of 7 units.”

Oh, but of course!

To answer the original question a different way:

Yes, a 3–4-6 right triangle is possible—in hyperbolic non-euclidean geometry, where the triangle is embedded in a surface of negative curvature such as a saddle shape. In addition the sum of the triangle’s 3 angles would be less than 180 degrees.

I agree with other remarks about how (in the Euclidean plane) a “staircase” connecting the vertices adjacent to the right angle can be constructed with steps of arbitrarily small size, whose total length nonetheless remains invariant at 7.