WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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In the next lesson, we are going to go over the triangle inequality theorem.
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Now, remember: inequality is when we are dealing with greater than/less than/greater than or equal to/less than or equal to.
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When we are talking about triangle inequality, we are talking about these; we are comparing different sides and angles together.
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So, the triangle inequality theorem says that the sum of the lengths of any two sides of a triangle is greater than the length of the third side,
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meaning (now, we have three sides in a triangle) the lengths of any two sides has to be greater than the third side.
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So, if I add up side AB with side AC, then this has to be greater than BC.
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And it just means, for example, that...let's say that point A is your home; point A is where you live--that is your home.
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Let's say you walk to school, and point B is school.
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Every day, you walk from point A to point B: you walk from home to school in the morning.
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And one day, you decide, "Oh, I want to stop by the market to get breakfast," or something.
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You want to stop by somewhere; let's say you want to stop by your friend's house to pick up your friend, so that you can walk together.
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This, let's say, is the market, or friend's house, or wherever.
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From home, you decide to go to the market, and then go to the school.
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The distance here, from home to school, is, let's say, x; now, if the distance from home to the market is y,
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and the distance from the market to school is z; if you are going to walk from home to the market, and then to school,
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you can agree that that is going to be a further distance than if you just walked from home directly to school.
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From home to the market to school is going to be greater than just from home to school; and that is what this inequality is saying.
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It is saying that if you are going to add up any two sides, that is going to be greater than just directly from a point to another point,
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because it is like you are detouring--think of a detour: whenever you detour, it is a further distance.
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So, again, from home to school is going to be a shorter distance, because it is direct; you are going straight to that point.
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It is going to be shorter; this alone is going to be less than home to market to school.
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Or you can say that home to the market, or home (point A) to point C, then to point B, is going to be greater than just going from point A to point B.
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That is the triangle inequality theorem: the length of two sides, if you add it up, is going to be greater than the third side by itself.
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Back to this right here: AB + AC is greater than BC, or I can say AB + BC is greater than AC.
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And the other two sides, AC + BC, are going to be greater than AB.
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You stop by a point; two of the lengths are going to be greater than the third length by itself.
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And that is the triangle inequality theorem.
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Here is a triangle, right here; and let's say that AC is 10, and BC is 8; I want to find possible lengths for AB.
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Now, I know that the sum of any two sides has to be greater than the third side.
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So, I know that this cannot be 1; why?--because if this is 1, well, then, 1 + 8 is 9, and then this is 10;
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so then, these added together have to be greater than the 10.
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So then, this can't be 1, because AB + BC will equal 9, and that is not greater than AC.
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It can't be 2, because if it is 2, then 2 + 8 is 10, and that is the same as AC.
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It can't be the same--you can't say that if you are going to walk from point A to point C,
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that is going to be exactly the same distance as if you walked from point A to B to C.
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So, you can't say that this is 2; now, can I say that it is 3?
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Yes, I can say that it is 3.
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Now, I can say that it is 3, because if this is 3, and this is 8, then that is 11, and that is greater than this.
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So, even though I can't say that it is 2, I can say that it has to be greater than 2.
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That means that it can't be 2, but it could be greater than 2.
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So, it could be 2.5, because 2.5 + 8 is 10.5, which is still greater than AC.
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So, it works as long as it is greater than 2...but there is a maximum number, too,
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because you are not just adding up these two sides to have it be greater than 10.
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It is just like the previous slide--remember: we went over AB + BC is greater than AC, and BC + AC is greater than AB, and then the other one.
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This can't be 20, because if I add up AC + BC, that is 18; and if this is 20, then you are saying that point A to point B
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is farther than if you go from point A to point C to point B, if you are saying that this is 20.
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Let's say that this is 20 miles; if this is 20 miles, then A to C to B can't be 18.
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So, we know that it has to be greater than 2; but then, if I add up AC and BC, then that becomes 18.
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So then, I know that this has to be shorter than 18, because if this your home to school,
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then it obviously has to be less than 18 miles; you are not going to be walking 18 miles to school...but anywhere;
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from point A to point B has to be a shorter distance than if you are going to go from point A to somewhere else, and then back to point B.
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So then, if this is 18, then this has to be shorter than 18; so then, it has to be less than 18.
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BC has to be greater than 2, but less than 18; those are the possible lengths of AB.
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Oh, wait; I wrote BC, and I need to write AB; so again, AB has to be greater than 2, but it has to be less than 18.
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So, it has to be any number between these two; and this is how you would write it.
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When you write an inequality, if you want to say that it is going to be between two numbers, you can say AB,
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or x, or whatever it is, has to be greater than 2, and less than 18; and that is how you state AB being between 2 numbers.
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How did we come up with these numbers, again?
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I know that AB + BC has to be greater than 10; so then, this has to be greater than 2, because 2 and 8 make 10.
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So then, it has to be greater than 2 so that this will be greater than 10.
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The same thing here: BC and AC together have to be greater than AB, so if they make 18, then this has to be less than 18.
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Those are two possible numbers that it can be between.
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Now, let's say that I give you three numbers; let's say this is 15; this is 7; and this is 6.
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And I ask you, "Can these three be the lengths of these sides?"
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Your answer is going to be yes or no, or true and false.
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Again, we have to see that the sum of any two sides has to be greater than the third side.
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What I can do here, to see if it can be true, is: I am going to add up the two smallest sides,
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because obviously, if this is the greatest side, if I add this to any one of these, it is going to be greater than the third.
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The biggest plus anything is going to be bigger than the smallest, and the biggest plus the other small one is going to be greater.
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We don't have to actually add up all pairs; instead, just add up the smallest lengths--that is 6 and 7.
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6 and 7 (AB + BC) should be greater than AC; AB is 6, plus BC is 7; is that greater than (this is a question mark) 15?
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13 (6 + 7 is 13)...is it greater than 15? No, it is not.
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So, this one is false, because if you add up 6 and 7, it makes 13; that is smaller than 15, so this one is "no."
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And it is just common sense; if you were to walk from point A to point C, then for you to stop by point B,
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before going to point C--it can't be that this is going to be shorter, or else you are going to always walk the long way!
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Again, this is false; this is not true.
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Let's go over some of our examples: the first one: Determine if the three numbers can represent the sides of a triangle.
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We have 6, 10, and 4; and we are going to see if these can be the lengths of the three sides of our triangle.
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And again, if I take the biggest one, and I add it up to one of the two numbers, we are obviously going to have a bigger number.
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So, we don't have to deal with the biggest number; instead, take the two smallest numbers, which are 6 and 4;
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add them up, and just see if they are going to be greater than the third one.
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So, 6 + 4 is 10; you know that 6 + 4 should be greater than 10, but is it?
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No, 10 is not greater than 10; so this one is "no"; it cannot be the three sides of a triangle.
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And the second one: 18 + 22 (those are the two smaller sides) should be greater than 45; 18 + 22 is 40.
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We know that 40 is not greater than 45, so then this one is also "no."
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These cannot be the sides of the triangle, the lengths of the triangle.
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If two of the measures of sides are given, between what two numbers must the measure of the third side fall?
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Again, we are given two sides, and we want to know possible measures for the third side.
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And we are going to write that as an inequality.
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18 and 22: if I have a triangle, and one side is 18, and the other side is 22, the let's call that third side x.
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What I can do is: I know that 18 + x has to be greater than 22.
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So then, x has to be (if I subtract 18) greater than 4.
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I also know that 22 + x has to be...
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or let's say 18 + 22 has to be greater than x; so this is going to be 40, so x has to be smaller than 40,
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because again, these two have to be greater than 22, and those two have to be greater than x.
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That means that x has to be greater than 4, but then x has to be smaller than 40.
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Now, sometimes this could be hard to read, because x is on the other side.
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Think of it as 40 > x, so then, let's do it by height; if Sarah is taller than Tim,
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then I have to say that Tim is shorter than Sarah.
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So, if 40 is greater than x, that means that x is less than, or smaller or shorter than, 40.
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So, if it helps, you can flip this around; it is less than 40.
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So then again, if you want to write between two numbers, you are going to write that x is going to be greater than 4.
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Remember: if 4 is smaller than x, then x has to be bigger than 4.
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And then, x is also less than 40--between 40 and 4.
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And just to show you a shortcut to find this: all you do is subtract the two numbers to get the smallest number possible...
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or not even the smallest; it just has to be greater than that number.
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And then, you add the two numbers, and it has to be smaller than that number.
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So then, x has to be bigger than when you subtract the two numbers, and it has to be smaller than when you add the two numbers.
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Here, x has to be greater than...we will subtract them and get 88.
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And then, you add them; it has to be 92, so it has to be smaller than 92.
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The same thing with D and E: x has to be greater than D - E, and then...
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Now, just in case E is, let's say, bigger than D, I am going to put this in absolute value.
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And then, it is less than D + E; and with this one, I don't have to put an absolute value, because there is no chance that it could be negative.
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If E is bigger than D, then if you subtract them, they are going to become negative--so it is just the absolute value, whatever their difference is.
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x has to be greater than that number, and then it has to be smaller than D + E.
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The next example: Indicate whether each statement is always true, sometimes true, or never true.
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Here we have, let's see, a quadrilateral; we have several different triangles: 1, 2, 3; and then, we have another one right here--4.
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We are going to see whether each one could be always true, sometimes true, or never true.
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BE: if this is 10, and this is 4, is BE going to be greater than 14?
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Well, if we look at this triangle right here, any two sides have to be greater than the third side.
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This is 10; this is 4; that is 14; that means that this has to be less than 14.
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But it says that it is greater than 14; so then, this one is never true, because,
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if this is, let's say, 20--it is saying BE has to be greater than 14; 20 is greater than 14--
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then you are saying that from point E to B is 20, and if you go to E to A, then to B, then it is 14.
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Isn't it true that, if you are going to go from this point directly to this point, that should be the fastest way,
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instead of having to cut, detour, and then go that way?
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This one is never true; this cannot be greater than 14.
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BC (the next one) is 13; BD is 5; well, it doesn't look like it could be 5, because it looks like it is going to be longer than 5.
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BD looks like it is going to be longer than BC.
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But do not assume that it is automatically going to be that way, just because the diagram looks like it.
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We are just going to base it on facts.
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Now, BC is 13; CD is 9; can BD be 5?
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Well, again, here is a triangle; you are going to add up the two shortest sides, so 5 + 9 is 14; is that greater than the third side, 13?
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Yes; that doesn't mean that it is going to be 5, but it could be 5.
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It could be, because by the triangle inequality theorem, just as long as two sides add up to be greater than the third side, it will work.
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But it doesn't mean that it is going to be 5.
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This one is sometimes true, because it can work, but that doesn't mean that it is always going to be 5.
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There are a lot of numbers that work; this is just one of them, so this is sometimes true.
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And the third one: If angle A is a right angle (let me erase these, because those numbers don't apply to the next one),
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then BD (this one right here) is going to be less than BA.
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OK, just to draw it out to maybe make it a little easier to see: angle A is a right angle--that is angle A;
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this is 10; this is 12, because it is the whole thing, if this is BD; BD is going to be less than BA.
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No; remember: last lesson, we went over the theorem that says that the side opposite the greatest angle is going to be the longest side.
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So, if this is a 90-degree angle, the hypotenuse is going to be the longest side of a right triangle.
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So obviously, BD cannot be less than AB or BA; so this is never true.
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The fourth example: We are going to determine whether it is possible to have a triangle with the given vertices.
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Is it possible for my triangle to be on these coordinates?
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And the way you figure that out is to find the distance between them and see if any of the two sides (the distance)...
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if the sum of the two shortest sides is greater than the longest side (then it will work).
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The distance formula, if you don't know, is (x₂ - x₁), or you could do (x₁ - x₂)--
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it doesn't matter--squared, plus (y₂ - y₁), squared; and again, don't think that this is x squared;
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and this is not y squared; it is saying the second x and the second y, the first x and the first y,
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because you are finding the distance between two points; so then, you are going to have (x₁,y₁) and (x₂,y₂).
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But it is not x squared; don't make that mistake--you are not squaring that number.
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Let's say the distance between A and B first: for AB, you are going to have 1 (x₁), minus -2, squared,
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plus 1 - 5, squared, equals...here this becomes a plus; that is 3 squared; that is 9, plus...that is -4 squared; that is 16,
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which equals the square root of 25, which is 5.
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Then, AC is going to be 1 - -5, squared, plus 1 - -4, squared.
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So then again, this is plus; so I have 6 squared (that is 36), plus...this is also plus...so that is 5 squared; that is 25.
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That is the square root of 61.
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And then, BC is the last one: for BC, it is -2 - -5, squared, plus 5 - -4...
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This becomes 3 squared (is 9), plus 9 squared (is 81); so here we have the square root of 90.
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OK, now, if you have a calculator, you can use your calculator to change this to a decimal.
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That way, you can compare these numbers with each other.
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If not, I can just kind of estimate what decimal this is going to be.
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If I know my perfect squares, I want to see what numbers this number falls between.
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Well, let's see: 7 times 7 is 49, so the square root of 49 is 7; and then, the square root of 64 is 8.
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It falls between those, but it falls...maybe a little bit between...right here, a little bit closer to the 64.
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So, this is probably going to be around 7.7 or 8, or something like that; this is around 7.8, let's say.
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This one is the same thing: 9 times 9 is 81, and then 10 times 10 is 100; it falls kind of right in the middle.
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If this is 9 and this is 10, something that falls in the middle is going to be around maybe 9.5.
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Notice how it is not an equals sign; it is squiggly--it is saying "approximately," "around 9.5."
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The two shortest sides would be 5 and 7.8; now, I know that, if I add these two together, for sure I am going to get greater than 9.5.
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So, 5 + 7.8 is about 12.8, and 12.8 is greater than 9.5; so this is "yes"; these vertices work; it is possible.
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That is it for this lesson; thank you for watching Educator.com.