Properties of shapes | Worked example

- [Instructor] We're told a triangle has side lengths of 3, 5, and X. Which of the following could be the value of X? And it says, choose all answers that apply. So pause this video and see if you can figure that out. Okay, now let's try to do it together. And there's different ways that you could approach this. You could just try to intuitively imagine a triangle that has side lengths 3, 5, and 3, and then another one that has side lengths 3, 5, and 5, and another one that has side lengths of 3, 5, and 9, and see if you can construct a triangle like that. But before we even do that, we'll do something like that. I just want to introduce or maybe review something that you might have seen before known as the triangle, triangle inequality. And hopefully you'll see that it's a pretty intuitive thing, but it actually does show up a lot in mathematics. And this is this idea that if you have a triangle, let me just draw an arbitrary triangle right over here. Let's say the length of this is is A. The length of this side is B. The length of this bottom side is C. All the triangle inequality tells us is that the longest side can not be longer than the sum of the other two. Or another way of thinking about it is it has to be less than or equal to the sum of the other two sides. Now why is that the case? Well, if you wanted the side of length C, if you wanted C to be as big as possible, what you'd want to do is spread open this angle as much as possible. But let's think about what happens. So you might spread it out even more and it might look like that to get C to be even longer. But the extreme case is where that this angle goes all the way to 180 degrees and it looks like this, where it's A and B. So the extreme case is when C is equal to A plus B. C can never get larger than A plus B. So C is always going to be less than or equal to A plus B. The reason why this is useful is because a triangle inequality tells us that you can never have, you can never, we can try out these numbers but if the longest of the numbers is longer or greater than the sum of the other two, well then we're going to be in trouble. So now that we have that out of the way, let's try to visualize these. So choice A would make X a 3. So we'd have a 3, a 5, and a 3. So our side lengths would be 3, 5, and 3. And we could try to visualize that. This could be side length 5, we could have a side length 3 like that, and then another side of length 3 like that. And it seems reasonable just visually, and it looks like I roughly drew it about 3, 5, 3. But also it meets the conditions for the triangle inequality. Our longest side, this side right over here, the 5, is less than or equal to the sum of the other two sides. 3 plus 3 is 6. 5 is for sure less than or equal to 6. So I like this choice. Now what about 5? Well that would imply a triangle that has side lengths 3, 5, and now X is going to be 5 again. We could try to visualize that. Let's say I have a side length 3 like that, and then I would have one side that is a 5, and then I have another side that is a 5. And it seems visually reasonable. And we can also validate that it meets the triangle inequality. Here, 3, this base isn't the longest side. One of the 5s would be the longest side. So let's just say that is our longest side, or it's one of the longest sides. Notice 5 is for sure less than or equal to 5 plus 3. It's for sure less than or equal to 8. And you could obviously do the same thing with this 5. It is for sure less than or equal to 5 plus 3. So it meets our conditions for the triangle inequality. And it also just makes sense when you draw it out like this. So that could be one of the choices. Now what about 9? Well that would be a triangle that has sides 3, 5, and X here would be 9. So immediately, your spider senses might be tingling a little bit. Because you might realize that, look, 9 is not less than or equal to the sum of 3 and 5. It is not less than or equal to 8, which is 3 plus 5. And you could see why not only does this violate the triangle inequality, why this would be an impossible triangle to construct. If I have a side length 9 right over here, and let's say I have a side length 3 right over here, no matter how hard I try, no matter how hard I try, the side length 5 is only going to be able to go about that far. And if I were to make this angle completely open wide, and then this point would essentially go down all the way here, the best I could do is something, and I'll do it in red, where this point right over here maybe gets to right over there, so that's 3, and then this distance right over here is 5, but we can only go from this point to this point a total distance of 8. We can't get all the way to this other point, which is 9 away from this original point. And that's what the whole point of the triangle inequality was. So this is not a possible triangle. So this is, we would just would not select this one.