Main content

# Construct a triangle with constraints

Here's a challenge: in this problem we are given constraints and asked to construct a triangle. It can be done! You'll learn about degenerate triangles, too. Created by Sal Khan.

## Want to join the conversation?

- 3:36-Degenerate triangle! I've never heard of this before, but I guess it would have to: two angles of zero, and one of 180; why is is called "degenerate", though?(38 votes)
- A degenerate case is a member of a certain class that is so different from the other members that it belongs to another, usually more simpler class. e.g. a triangle formed by three points all of which are on a straight line is better considered as a line segment. These are the limits of the definition of things. The degenerate triangle satisfies the definition of a triangle (it is a triangle) but it is better/simpler to consider it a line segment.(42 votes)

- I still don't understand what a degenerate triangle is?(11 votes)
- it is the triangle one obtains when one angle has a value of 180 degrees and the other angles are zero.(9 votes)

- Who walk up to you and ask you that?(11 votes)
- Maybe a math teacher?(8 votes)

- In mathematical terms, a constraint is a predefined condition that has to be met. It is called a constraint because it constrains, i.e. limits or restrains, the many ways you could tackle something. It applies to everything, not just triangles. In the video, Sal shows how to draw a triangle - however, not just any triangle, but a triangle that has to meet certain requirements (constraints), for example, it has to have the side lengths 3, 3, and 5. These are constraints that the triangle has to meet.

There might be constraints that make it impossible to build the triangle (such as side lengths 2,2, and 5) or constraints that only yield one unique way of building the triangle (see the last triangle with side lengths 3,3, and 5). A constraint could also be to draw a triangle that has a right angle.(7 votes)- All of the sides of a triangle has to = 180 degrees(6 votes)

- I have no clue what i just watched(7 votes)
- why why why why why why why why(5 votes)
- What is a line segment?(3 votes)
- A line segment is a geometric figure that consists of two distinct points, called endpoints, and all the points straight between the two endpoints.

Note that a line segment has finite length, though it contains infinitely many points.(5 votes)

- I seem to not understand the questions(4 votes)
- This is not what I'm looking for, this is the question I cannot figure out:

Q1. Construct a right-angled triangle containing an angle A such that sin A = 0.4.

Help me?(3 votes)

## Video transcript

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.