8th grade foundations (Eureka Math/EngageNY)
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.
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- 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?(31 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.(34 votes)
- I still don't understand what a degenerate triangle is?(8 votes)
- it is the triangle one obtains when one angle has a value of 180 degrees and the other angles are zero.(5 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.(6 votes)
- All of the sides of a triangle has to = 180 degrees(5 votes)
- Who walk up to you and ask you that?(6 votes)
- Maybe a math teacher?(2 votes)
- What is a line segment?(2 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)
- 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)
- I have no clue what i just watched(3 votes)
- i can tell that the triangle at1:00could never really work but it still doesn't really make cense can someone help a sister out(2 votes)
- they have to be able to reach, that's all
it's not a triangle yet if one of the corners is left standing open or if one of the lines sticks out on it's own?
so the sides have to be long enough so any two of the sides added together equals longer than the leftover side
just like all the angles in a triangle have to add up to 180 degrees so the corners can meet
otherwise the three lines can't close to make the triangle
it's so simple it's gorgeous
this is something you will always be able to rely upon about triangles, and why triangles are used for measuring and navigation and why there's such a thing as trigonometry which is...
the ends have to reach
- What is the difference between a degenerate triangle and a line segment? Or is there any difference?(3 votes)
- a degenerate triangle is just a triangle with no area i think(1 vote)
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.