Constructing triangles with constraints

If someone walks up to you
on the street and says,

all right, I have a
challenge for you.

I want to construct a triangle
that has sides of length 2.

So sides of length-- let me
write this a little bit neater.

Sides of length 2, 2, and 5.

Can you do this?

Well, let's try to do it.

And we'll start with the longest
side, the side of length 5.

So the side of length 5.

That's that side
right over there.

And now, let's try to draw
the sides of length 2.

Every side on a
triangle, obviously,

connects with every other side.

So that's one side of length 2.

And then this is another
side of lengths 2.

Another side of length 2.

And you might say, fine, these
aren't touching right now,

these two points.

In order to make a triangle,
we have to touch them.

So let me move them
closer to each other.

But we have to remember, we
have to keep these side lengths

the same.

And we have to keep touching
the side of length 5

at its endpoint.

So we could try to move them in.

We could try to move them in,
but what's going to happen?

Well, you could rotate
them all the way down

and they're still not going
to touch because 2 plus 2

is still not equal to 5.

They rotate all the
way down, they're

still going to be 1 apart.

So you cannot construct
this triangle.

You cannot construct
this triangle.

And I think you're noticing
a property of triangles.

The longest side cannot
be longer than the sum

of the other two sides.

Here, the sum of the
other two sides is 4.

2 plus 2 is 4.

And the other side is longer.

And even if the other
side was exactly

equal to the sum of
the other two sides,

you're going to have
a degenerate triangle.

Let me draw that.

So this would be side,
say, 2, 2, and 4.

So let's draw the
side of length 4.

Side of length 4.

Side of length 4.

Let me draw it a
little bit shorter.

So that's your side of length 4.

And then, in order to make the
two sides of length 2 touch,

in order to make them touch,
you have to rotate them

all the way inward You
have to rotate them

all the way inward so that
both this angle and this angle

essentially have to
become 0 degrees.

And so your resulting triangle,
if you rotate this one

all the way in and you
rotate this all the way in,

the points will actually touch.

But this triangle will
have no area anymore.

This will become a
degenerate triangle.

And it really looks more
like a line segment.

So let me write that down.

This is a degenerate.

In order for you to draw
a non-degenerate triangle,

the sum of the
other two sides have

to be longer than
the longest side.

So for example, you could
definitely draw a triangle

with sides of
length 3, 3, and 5.

So if that's the side of
length 5, and then this--

if you were to rotate
all the way in,

those two points would-- let me
draw this a little bit neater.

So let's say that's
where they connect.

And we know that
we could do that,

because if you think about it,
if you were to keep rotating

these, they're going to pass
each other at some point.

They're going to
have to overlap.

If you tried to make
a degenerate triangle,

these points wouldn't touch.

They'd actually overlap by
one unit right over here.

So you could rotate them
out and actually form

a non-degenerate triangle.

So this one, you
absolutely could.

And then there's another
interesting question,

is this the only triangle
that you could construct

that has sides of
length 3, 3, and 5?

Well, you can't
change this length.

So you can't change that
point and that point.

And then, you can't
change these two lengths.

So the only place
where they will

be able to touch each other is
going to be right over there.

So this right over here
is the only triangle

that meets those constraints.

You could rotate it
and whatever else.

But if you rotate this, it's
still the same triangle.

This is the only triangle
that has sides of length 3, 3,

and 5.

You can't change
any of the angles

somehow to get a
different triangle.