Trapezoid diagonals conjecture

0:00- [Instructor] What we're gonna do in this video is think a little bit more about trapezoids, but in particular, I'm gonna think about the diagonals of trapezoids and see if we can come up with any properties that could be interesting. So if, let me just draw a trapezoid here, and I'm just going to try to hand draw it the best that I can.

0:20So let's just assume that this is a trapezoid. And let's think about its diagonals. So that is one diagonal, and this is the other diagonal. And my question is, do we think that these diagonals are going to have the same length? And I encourage you to pause this video and think about, is that going to be true?

0:41Is it going to be true for any trapezoid? The way I drew it just now, it looks like they could have the same length, but let's see if I could make a trapezoid that could have a very different, that where the diagonals clearly have different lengths. What if I had a trapezoid that looked like this?

1:00So maybe the top is much shorter than the base or than the bottom base. It looks something like this. So in this trapezoid right over here, this diagonal that I'm drawing is clearly shorter than that diagonal. So we know that the diagonals are not congruent for any trapezoid.

1:26We can see that just by inspection. But I still keep coming back to the one that I drew the first time. And so, well, this one looks like it might work. Maybe if we're dealing with an isosceles trapezoid, maybe that would do the trick. Maybe in that situation, I have congruent diagonals.

1:51And so what I just said there, that is known as a conjecture, conjecture, which in math is really a theory, a hypothesis of something that might be true. And you need to have a conjecture in order then to try to prove to yourself whether or not that's true. In another video, we'll actually try to do a proof.

2:11But let's see, does this conjecture, isosceles trapezoids, could we expand that? Or is it only for certain definitions of trapezoids? For example, there is an inclusive definition of trapezoids and there is an exclusive definition. The inclusive definition says anything that has at least one pair of parallel sides is a trapezoid.

2:34So by the inclusive definition, inclusive definition, the thing I drew in the beginning, both of these are trapezoids, but a parallelogram would also be a trapezoid. So by the inclusive definition, something like this would also be a trapezoid where it definitely has one pair of parallel sides, but it actually has two pairs of parallel sides.

2:58And now let's see if the diagonals here look congruent. Well, if I look at this diagonal and this diagonal, it looks pretty clear that this one is shorter. So we are not congruent in that situation. And this one would be isosceles in the tradition, if we're taking the inclusive sense of it, where this side is equal to that side.

3:23But let's think about, so, I wouldn't wanna use the inclusive definition of the trapezoid to say isosceles trapezoids have congruent diagonals. But in the exclusive, in the exclusive definition of a trapezoid, where you say you have exactly one, exactly one pair of parallel sides, you can't have two, then you are dealing.

3:44And if we're saying for that definition, for the exclusive definition of trapezoids, definition of trapezoids, isosceles trapezoids have congruent diagonals. This feels like a pretty interesting, interesting conjecture that maybe we will prove in another video.

4:16And actually, I do prove that in another video.