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Triangle Side Inequalities - Concept
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The **triangle inequality** theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. To find a range of values for the third side when given two lengths, write two inequalities: one inequality that assumes the larger value given is the longest side in the triangle and one inequality that assumes that the third side is the longest side in the triangle. Combine the two inequalities for the final answer.

Let's say I gave you 3 pieces of spaghetti and you try to make a triangle out of them, so I'm going to say that this piece of spaghetti is 10 inches long, this piece right here is 3 and this piece right here is 6. Does it look like we're going to be able to make a triangle out of this? It looks like we don't quite have enough to close that arch, let's say however instead of 3 and 6 let's say I gave you pieces that were 3 and 7, then we would just have a straight line. Because there's no way for us to form an angle there.

What we're talking about here are triangle inequalities which means if you look at a triangle there's a special relationship between those 3 sides and it kind of make sense if you think about it. If you started at point a what's going to be the fastest way to go to point c? Is it going to be to walk to point b and then point c? or is it just going to be faster to go from point a to point c? Well pretty clearly you're just going to walk straight to point c. But how does this look in terms of relationships? Well we can say that line segment ab or side ab plus side bc must be greater than your third side ac, so what's that saying is that going straight from one point to another is going to be shorter or less than the sum of the other 2 sides. You can write 2 other inequalities, we could start with ab and go with our side ac and say ac has to be greater than your third side which is bc. At last you could write your last inequality which we'd say side ac plus side bc has to be greater than your side ab. So in essence you always have to make sure that two of your sides summed will be more than the third side which is why we couldn't make a triangle because 3 plus 6 is 9 which is less than 10. 3+7=10 but notice you do not have an equal to part to your inequality all you have is this greater than symbol.

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