 This video is called the range. You're going to be asked in your book work and on tests, and expected to know how to find the range for the measure of the third side of a triangle, if you're given the measures of two sides. We just learned in the previous video that you need to take the sum of the smaller two sides, and that needs to be bigger than the third side in order to make a triangle. This is just a little bit different, but it's the same type of thinking. It's asking if we have a triangle ABC, and we're told that AB is nine units long, and AC is seven units long, it's asking us, well, what could this third side be, and we'd still have a triangle? So if you look at it as it is right now, you can see the third side is 5.85 units long, and sure enough, that looks like a triangle to me. But let's start playing with it. ABC, do I still have a triangle if CB is at 2.49? I sure do. Do I have a triangle if C is way over here, where BC is 13.27 units long? I sure do. It still looks like a triangle. So notice how AB hasn't changed its state at nine, AC hasn't changed its state at seven, but we're playing around with this third side. Is it still a triangle out here? When we're at 15.7, 15.8, we still have a triangle. So it's asking us, what is the range? How small can that third side get and still be a triangle? And how big can that third side get and still be a triangle? So let's take a look. That third side is getting smaller and smaller and smaller, but it's still a triangle. At what point did it stop becoming a triangle? It's right here at 2, because now at 2 it's a straight line. If you keep going, it becomes a triangle again, but notice what happens. 2 is as small as it gets. It starts going up the length, and the length starts going up again. So it looks like if our third side is 2, we're not going to have a triangle, but if it's just above 2, we will. So our third side has to be greater than 2. And then let's see what happens on the other side when we make it really big. Our third side is getting bigger and bigger. 12, 13, 14, 15. Here we go. It hits 16. At 16 it became a straight line. I lost the triangle, and notice it started going down again. 15, 14. So it looks like when I hit 16, that's the biggest I can get, and it does not make a triangle. So it looks like my triangle, my third side has to be less than 16 because that makes it a triangle, but it has to be bigger than 2. Oh, sorry, I don't know what happened there. Hold on. All right, sorry about that. So let's go ahead and look at your actual note sheet. And the first example says, find the range of the measures of the third side of a triangle given the measures of the two sides. So we just did an example. Looks like it was down... Looks like it was down here where our side lengths were 7 and 9, and we said that it had to be bigger than 2, but less than 16. So you could write that as your third side if we called it n, it would have to be larger than 2, but less than 16. So that would be the answer. Let's try our next one. Let's go ahead and do... I have these all goofed up. I apologize. Let's go ahead and fill in the one for when the two side lengths are 5 and 8. Let's go back and look at our picture. So if I change one side length to 8 and the other side length to 5, so now I have AB is 8 and AC is 5. Let's find the range. Let's figure out what could that third side be and have a triangle. So we'll make it little. It works when it's in the 4s. When it's in the 3s, it looks like when it hits 3 is when we get our straight line and that is the smallest we'll go because if I keep going this way, the number starts getting bigger again. So it looks like I'm going to have to be bigger than 3, but now let's look on the other side. It's getting bigger 12. When it hits 13, it hits that straight line and that's the biggest it will get because it'll start going back down again. So it looks like I have a triangle if the side length is less than 13 but bigger than 3. So let's go ahead and fill that in. So for 5 and 8, I need what was it? Bigger than 3, but less than 13. So I would write this as n is greater than 5, but less, I'm sorry, not greater than 5, greater than 3, but less than 13. So that would be my range. That's saying, that's simply saying that when I have two triangle side lengths of 5 and 8, my third one could be 4, 5, 6, 7, 8, 9, 10, 11, 12, all the way up to really 12.999 repeating. Now, you don't have access to geojibra all the time and you don't get to play with the triangles like that all the time, so is there a different way we could solve this where you don't need the picture? Is anyone catching a pattern? Look at the first example. When you add 9 plus 7, you get 16. When you subtract 9 minus 7, you get 2. Over here, when you add 5 plus 8, you get 13. When you do 8 minus 5, you get 3. So if you follow this pattern to do our last example of 12 and 18, if we add 12 and 18, we get 30, and then if we subtract 18 minus 12, we get 6. So that would tell me in the range that third side has to be bigger than 6 but less than 30. Now, if you use this trick of just adding the numbers together and subtracting the numbers, that's great. You'll get the right answer, but just keep in mind when you do the subtracting, always do the biggest bigger number minus the smaller number so you get a positive answer. You can't have a negative side length. Alright, so good luck with your range problems.