Kites as a mathematical shape - Engineering Mathematics Video Lecture

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 kites 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.