 In this video, I'm going to talk about verifying conjectures, verifying conjectures using the law of detachment. So basically what this is, is an example using the law of detachment. So determine if each conjecture is valid by the law of detachment. Now remember our law of detachment uses conditional statements to figure out if something is true or not. So we have this, given if the side lengths of a triangle are 5 centimeters, 12 centimeters, and 13 centimeters, then the area of the triangle is 30 meters squared. So now think about that statement for a moment. We're talking about the area of a triangle. Area of a triangle is 1 half base times height, 1 half base times height. So you take half of the base and half of the base and then times the height to figure out what the area is. They said that the area was 30, that makes sense. So now they're also saying the area of the triangle PQR is 30 centimeters squared. The conjecture is, now this is what we're determining if this is valid. This is the part where we're trying to determine if it's valid. Conjecture is the side lengths of the triangle PQR are 5 centimeters, 12 centimeters, and 13 centimeters. So what we've got to do is we've got to see if that's valid. Is that true, is that a valid conjecture, is that a valid conclusion? Now when you think about a little bit, we know that the area of a triangle is 1 half base times height. So you take the base and you take the height, you multiply them together and then you divide by 2. Now in that process what they did is they took the base, the height, multiplied it together, divided by 2 and they got 30 centimeters squared for this triangle PQR. So does that really mean, does that really mean that the triangle has to have sides of 5, 12, and 13? Okay, if we have a triangle, if we have a triangle, sides of 5, sides of 12, and sides of 13, and they have to be in this order, they have to be in this order because 5 and 12 are the two smaller legs and 13 is the hypotenuse. A little extra here, this is actually a Pythagorean triple. If you want to look that up, that's a Pythagorean triple, so I know it's a right triangle, but anyway, 5 and 12 are, 5 is the height, 12 is the base, and so if you take 5 times 12, you get 60, 60 divided by 2 is 30. So that's logical. That does make sense that, okay, 5 and 12 are your side lengths, your height and your base that you need to get a area of 30. That makes sense, not a big deal. But they're saying down here, they're concluding since the area of the triangle is 30 centimeters squared, they're saying the side lengths of the triangle have to be 5, 12, and 13 centimeters. Well, if you think about that a little bit, if I have a triangle, if I have a triangle, I could actually come up with different ways, different side lengths other than 5 and 12 to get me an area of 30. So, for example, I might have a side length of, we'll say, 6 and 10. I don't know right off the top of my head what this side length would be. I don't know exactly what this side length would be, but if I have these two side lengths of 6 and 10, 6 times 10, base times height, if I take 6 times 10, that's 60, 60 divided by 2 is 30. So this also has an area of 30 centimeters squared. This area is also 30 centimeters squared. So that kind of, this conjecture is not really valid because I just came up with another example of a triangle with an area of 30 centimeters squared, but it doesn't have sides of 5, 12, and 13. So this is not valid. This is actually a not valid statement. Now, why is it not valid? Well, I came up with what's called a counter example. I came up with a different example where, yeah, the area is 30 centimeters squared, but the side lengths are not 5, 12, and 13. My side lengths are actually different, 6, 10, and then whatever the third side would be. The third side right now is irrelevant because it's not going to help us find the area. So that's an example of verifying to see if a conjecture is actually valid. This one, in fact, was not valid because I was able to come up with a different example. I was able to come up with a different example. Okay, so that's one example of using the law of detachment. Here's another example. Here's another example. So we're doing the same thing, determine if each conjecture is valid by the law of detachment. So given, all right, in the World Series, if a team wins four games, then the team wins the series. Okay, so notice that I got a conditional statement. If, hypothesis, then conclusion. So if a team wins four games, then the team wins the series, okay? The Red Sox won four games in the 2004 World Series. What conclusion can you make from that? Okay, the Red Sox, they won four games in the World Series. One might make the conjecture, the conclusion, that the Red Sox won the 2004 World Series. Okay, so let's see if that makes sense. If a team wins four games, then the team wins the series. The Red Sox won four games, the Red Sox won the World Series. That is absolutely valid. That is absolutely a valid statement. Okay, so notice how everything followed a logical point of view. Okay, so notice how everything followed a logical progression. If a team wins four games, we can call that P, that's the hypothesis. Then the team wins the series, we can call that Q. The Red Sox won four games in the World Series. Again, that's P, that's winning four games. That's the hypothesis, the same hypothesis up here. Then the Red Sox won the World Series. That's the same conclusion. They won the series. That's the same conclusion. So notice how we go from P to Q, and then from P to Q. That's using the law of detachment to see if this is valid or not. So let's actually compare that. Let's go backwards. Let's compare that to the previous example. If the side lengths of a triangle are 5, 12, and 13, so there's the P. There is our hypothesis. If the side lengths are 5, 12, and 13, then the area of the triangle is 30 meters squared. So there's the Q again. So there's our conclusion. Then the next they say, the area of the triangle PQR is 30 centimeter squared. The area, if I look back up here, the area was the conclusion part. Then this conjecture is the side lengths of the triangle PQR are 5, 12, and 13. That's the hypothesis. But notice the order in which everything went in. The first of all, if P and Q, P first, the hypothesis first, then the conclusion, this one, on the other hand, was the conclusion first and then the hypothesis. And that made it not valid. So that's one way of determining whether a statement is valid or not is determining what order, the hypothesis, and the conclusion come in. This one was P than Q, and then all of a sudden we went to Q than P. So this one was hypothesis, conclusion, and then all of a sudden we went conclusion, hypothesis. So notice the order in which we did this was not quite correct. So again, this was a not valid statement along with this example that I have here, the example that we came up with. So notice over here for the World Series and the Red Sox, P than Q, P than Q follows a logical order in progressions. That was a valid statement. All right, that was a couple of examples of how to verify conjectures using the law of detachment.