Topic B: Constructing triangles
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.