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Triangle Inequality Theorem and Examples

In mathematics, the triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side, if the triangle is not isosceles. The inequality is a consequence of the fact that the two shorter sides of a triangle must be adjacent (that is, one side of the triangle must be between the other two sides).

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Hi, Mr Kleinberg here with you. Another lesson here. We're gonna focus on the triangle inequality. The're, um and what this says or what this pertains to is a very specific relationship between the three sides of a triangle. You can't just throw out three lengths and create a triangle. There are specifics in terms of how three lengths can relate to each other in order to create physically a triangle. So what the triangle inequality Theorem says, is this. Given a triangle side length A, B and C, any two sides of some of them must be greater than the third. So, for example, a plus B must be greater than see B plus C must be greater than a and A plus. C must be greater than be. All three of those inequalities must be met. So if we look at an example, let's say this is 10 units and I have some length of eight units in some length of two units. And the question is, can I make a triangle? Well, if you think about it, if I were to swing this side down, it's gonna land at eight units. This is going to swing down and land at two units. Well, eight plus two is 10. You're gonna have a triangle that looks like this, and that's not a triangle. In the case of having three sides. In the case of context, you might call Call it a degenerate triangle. But if we're calling a triangle a polygon, a plainer shape that has three sides, this is not a triangle. So you can see how I have to have at least eight or to be bigger so that there's some is larger than 10. So there's a couple of questions you could be asked about. This one is I could give you a triangle with sides and say, Is this a triangle? You tell me and or or another way of saying it would be? Do these three side length determined a triangle so I could say something like 11, 2 and five, and I would say No. This is not a triangle. Two plus five is not greater than 11, even though 11 and to our greater than five. And even though 11 and five are greater than two, two plus five is integrated than 11. This would not make a triangle, so the triangle inequality here, Um, says you can't have a triangle with sides length of 25 11. I didn't have to give you the picture. I could have just written out to five and 11 and said, Hey, are you going to get a triangle out of this? And what you need to do is try every possible some of to and make sure that that's larger than the third side here, Clearly two plus five. Is that right? The beloved. Okay, so here's another example. I could say, um, does a triangle with sides 11, 21 16. So let's say I had 11 centimeters, 21 centimeters and 16 centimeter. Let's say I had those 33 segment lengths. Can we create a triangle? Well, is 11 plus 21 greater than 16? Yeah, because 21 by itself is greater than 16. How about this? 11 plus 16? Is that greater than 21? Sure, you're going to get 27 which is definitely bigger than 21 check and then the third combination is going to be 16 plus 21. Is that greater than 11? Well, again, both of these are bigger than 11 so check. So because we've met all three inequalities, we do have a viable triangle. Now, what I could say is all right if I give you two sides of a triangle, let's say one side is three and the other side is seven. My question is, what is the range of side lengths that the third side could be? So, in other words, what? What integers is see going to be between in order to satisfy the triangle inequality there? Um, well, think about it. Three plus seven has to be larger than the third side. Okay, so if you think all right, well, three plus seven equals 10. Then we can't have a side length that exceeds 10. In fact, we can't have a side length that equals 10. So our upper bound is going to be 10, assuming three and seven are the two smaller sites. Okay, so the other way of thinking it is one of three and seven aren't the two smaller sides, but one of them is the biggest side. Let's say seven is the biggest side. Well, then, if you look at this way and say seven minus three, which is four, if you think about this lower range of 44 plus three would be seven. So we have to be bigger than four in order for the inequality theory. But it's still work, so the range of sides that C could be is between four and 10. And so shortcut here is to find your range of values. You're going to find the sum and the difference of the two given sides, and your third side is going to be within those two values, not inclusive notice. I don't have less than or equal to. I just have less than symbols. Um, because you can't be equal to because it's an inequality. The're, um not equality theory. So those were some possible examples and questions of how to use the triangle inequality. Fear, Um, and what? The triangle inequality. The're, um, is

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Top Geometry Educators
Lily A.

Johns Hopkins University

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Oregon State University

Kristen K.

University of Michigan - Ann Arbor

Joseph L.

Boston College

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