Les cerfs-volants

In everyday language, we know what a kite means. It's these flimsy things that we take to the beach to fly in the wind with our families. But you could imagine mathematicians have looked at the general shape of these kites, or at least the way they're drawn in cartoons, and say, well, that's an interesting shape in its own right. Let's also make this a mathematical term. This is a shape like a parallelogram or like a rhombus. It's just another type of quadrilateral. But in order for it to be used in mathematics in a useful way, we have to define it a little bit more precisely. So let's see if we can come up with a couple of interesting definitions of what a kite could be or a couple of interesting ways to construct a kite. Well, one way that you could think about a kite is it looks like it has two pairs of sides that are congruent to each other. So, for example, it looks like this side and this side need to be congruent to each other. So let's make that a constraint. And they touch each other. They have a shared common endpoint. So you have one pair of congruent sides that's adjacent to each other. They have a common endpoint. And then you have another pair of sides that are congruent to each other. And they are adjacent. They share a common endpoint. So one definition that you could make for a kite is that you have two pairs of congruent sides, where the congruent sides are adjacent. And you might say, well, what's the other alternative? If the congruent sides aren't adjacent, what else could they be? Well, the congruent sides could be opposite each other. And what happens if you were to do that? So if these two sides are congruent, but they didn't have a common endpoint, we're still dealing with a quadrilateral. What would it look like? Well, you would have one congruent site here, and that would be congruent to this side right over here. And then you would have a congruent side right over here that is congruent to this side. This would be a situation where you have two pairs of congruent sides, but they're not adjacent. They don't have any common endpoints with each other. Each side in the congruent side pair, they're opposite to each other. So here, once again, we get a quadrilateral. We still get four sides. A kite is a quadrilateral. This is a quadrilateral. But this isn't a kite. This right over here is a parallelogram, and we've seen that multiple times before. But kites can also be constructed in other interesting ways. You might see that what looks right here, that these two diagonals of this kite are perpendicular. And that indeed-- and I'm not going to prove it here-- is a property of a kite. These two lines, these two diagonals, intersect at a 90-degree angle. The other thing we know about kites is that one of these lines is bisecting the other of the two. So you could actually construct a kite that way. You could start with a line, and then you could construct a perpendicular bisector of that line, another segment that bisects it at a 90-degree angle. So here, there you go. So that bisects it, so that means that this segment is equal to this segment. We split it in two. And then if you connect the endpoints of the segments, you should get a kite. And you will indeed get a kite. So it would look something like this. And once again, this segment is congruent to this adjacent segment, and this segment is congruent to this adjacent segment. But what would happen if these two diagonals are both perpendicular bisectors of each other? So what would happen in this scenario, where-- let me draw one segment. And then I'm going to make another segment, but they're going to be perpendicular bisectors of each other. So let's do that. So now they're both perpendicular bisectors of each other. So this segment is equal to this segment, and this segment is equal to this segment. Well, now, once again, you still have a kite, but now you're also satisfying the constraint for another type of quadrilateral that we've seen. So now you're satisfying the constraint. All your sides are equal. All of your sides are parallel. You're now dealing with a rhombus, which is also a special type of parallelogram. And then if you were to go even further, where these two diagonals have the exact same length and they're both perpendicular bisectors of each other, so you have both the exact same length. I'll try to draw it as cleanly as I can. So they're both the exact same length, and they're both perpendicular bisectors of each other. So each of these halves would be the same length as well. Then you have a subset of-- I guess I could say-- rhombi, and you get to a square. So one way of thinking about it is any square is also a rhombus. And any rhombus is also going to satisfy your constraints for being a kite. But there's a bunch of types that don't satisfy your constraints of being a rhombus or a square. A kite is just two pairs of congruent sides that are adjacent to each other, and they're usually pretty easy to spot out because they look like kites.