Kites as a geometric shape

in everyday language we know what a kite means is 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 that they've draw they're drawn in cartoons is it 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 have to be used in mathematics in a useful way we have to define it a little bit more price 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 as common endpoint and then you have another pair of sides you have another pair of sides that are congruent to each other they are congruent to each other and they are adjacent they share a common endpoint so one definition that you could make for kite is that you have two pairs of congruent congruent sides where the congruent sides wear congruent sides are adjacent congruent sides are add jacent 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 weren't if they didn't have a common endpoint and we're still dealing with a quadrilateral what would it look like well you would have one congruent side here and that would be congruent to this side right over here 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 and they don't have any common endpoints with each other each each each side in the congruent side pair they are 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 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 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 this right over here is a perp this is a purpose 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 you can 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 it would look something something like 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 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 you still have a kite but now you're also satisfying the constraint for another type of quadrilateral that we've seen so now use that you're satisfying the constraint all your sides are equal all of your sides are parallel you are 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 there did both the exact same length and 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 you get to a square you get to a square so one way of thinking about it is any square is also a rhombus which is also going to and any rhombus is also going to satisfy your constraints for being a kite but there's a bunch of kites that don't satisfy your constraints of being a rhombus of or square a kite is just two pairs of congruent sides that are adjacent to each other and they're usually pretty obvious pot out because they look like kites