00:00Hello, in this video, I invite you to review the entire course on the Pythagorean Theorem.
00:11The purpose of this video is to remind you and explain the most important elements of this chapter.
00:17More specifically, we will of course talk about the Pythagorean equality, but also its applications.
00:23To prepare for a test or even an exam, this will not be enough.
00:27You will still need to practice by doing many exercises for the course.
00:31Here we go.
00:34So, we cannot talk about the Pythagorean Theorem without talking about Pythagoras, Mr. Pythagoras, Pythagoras of Samos.
00:41In reality, it may not be Mr. Pythagoras, but gentlemen.
00:45We are not sure whether Pythagoras himself existed as a human being, but it could be that it was rather a school,
00:51which was called the Pythagorean School, which brought together many scientists, researchers, and philosophers.
00:59So, it is currently in southern Italy, but at the time, it was in ancient Greece.
01:04And so it is this school that would have formalized the Pythagorean Theorem because in fact the Pythagorean Theorem was already known well before him.
01:11We are certain of that, we have traces of it in particular among the Chinese or the Babylonians, so about 1,000 years before Pythagoras.
01:22But Pythagoras would have stated things in a more formal way, a bit like today, but of course with the writing of the time.
01:29So, what does the Pythagorean Theorem tell us?
01:32The Pythagorean Equality?
01:33Well, it is a relationship on the lengths of the sides of a right triangle.
01:39Here we see, we have drawn a right triangle and we have indicated its dimensions.
01:44The two sides of the right angle are 3 and 4.
01:47So 3 and 4, whatever the unit, centimeters if you like.
01:52And the long side, the hypotenuse, measures 5.
01:56Well, we're going to quickly discover what the Pythagorean Theorem tells us.
02:00It tells us that in a right triangle, you're going to take the long side alone and you're going to square the length.
02:12Which means that I'm going to do 5 squared and 5 squared here.
02:16That's 5 by 5, that gives me 25.
02:19And I'm going to take the other two sides, the sides that form the right angle.
02:23And I'm going to do the same thing, I'm going to take 3, I'm going to square it.
02:303 squared is x 3 equals 9.
02:33And I'm going to do the same thing for 4.
02:364 because 4 times 4 equals 16.
02:39But that's not all.
02:41I'm then going to take these two squares and I'm going to add them together.
02:44So I'm going to take my 3 squared which is 9 and my 16 and I'm going to add them together.
02:549 plus 16 25.
02:56Well, there you have our Pythagorean Theorem.
02:58I find 25.
03:03The Pythagorean Theorem actually tells us that if you take the square of the hypotenuse and you take the sum of the squares of the other two sides,
03:11well, you find the same number provided that the triangle is a right triangle.
03:17That's what we found here.
03:18And the Egyptians also knew the Pythagorean Theorem, particularly in its version 3-4-5 which is a particular version.
03:29Here, I showed it for a triangle whose sides measure 3, 4 and 5.
03:33But we can find the same equality with other sides, of different lengths.
03:38But the Egyptians liked the side of the triangle 3-4-5 because 3-4-5 is also three consecutive integers and they made a rope that was called a 13-knot rope where 13 knots were tied regularly,
03:54with the same space between the knots.
03:57And once we brought the two ends of our rope together, we could make a triangle, a 3-4-5 triangle as we see here.
04:11And it was very practical because if we had a very long rope, we could build very, very large right angles, which were useful for building walls and buildings.
04:20We can imagine that we weren't going to carry a giant set square to build right angles.
04:29But here, we rolled the rope in the cart and it didn't take up any space.
04:34And so it was extremely practical and it should be noted that this rope would still be used by masons in the 20th century to ensure that walls were perpendicular.
04:44But let's get back to our Pythagorean Theorem.
04:46So finally, the Pythagorean Theorem tells us that if we have a right triangle, well, the square of the hypotenuse here,
04:58the square of the hypotenuse is equal to the sum of the squares of the other two sides.
05:06We can also write it in algebraic form.
05:09If we have a right triangle whose sides measure a, b and c, well, in this case, we can write that a squared is equal to b squared plus c squared.
05:22A squared, the square of the hypotenuse is equal to the sum of the squares of the other two sides, that is to say b squared plus c squared.
05:29This equality is very simple to understand and very easy to remember.
05:36And what's more, as we'll see right away, it has applications.
05:41Using this equality, we can calculate lengths in a right triangle.
05:45Because if I know the length of two sides of my right triangle, then thanks to the Pythagorean Theorem, I can deduce the length of the third side.
05:57We'll see this in a very quick example.
06:01Without going into too much detail about the wording, this is the subject of other videos that I invite you to watch if you haven't already.
06:08Here, we're simply going to understand the principle for calculating lengths in a right triangle using Pythagoras.
06:17So, here's a figure where we're given the lengths of two sides.
06:21So a, b is 6 centimeters this time and a, c is 9 centimeters.
06:25We're then going to interpret the Pythagorean Theorem in this situation.
06:30The Pythagorean Theorem tells us that if triangle a, b, c is right angled at a, it is right angled at a.
06:36And in this case, we will have b, c squared the square of the hypotenuse equals a, b because plus a, c because the sum of the squares of the other two sides.
06:51Which means that here d, c squared will be equal to a, b squared or 6 squared plus a, c squared or 9 squared.
06:57So, 6 squared and 9 squared can be calculated.
07:06This means that b, c squared is equal to 36 squared 6 plus 81 squared 9.
07:12We can also add them together.
07:16This equals 117.
07:18Which means that b, c squared equals 117.
07:21Now, let me remind you that the length b, c is not given to me.
07:25We can clearly see it here.
07:27There is a question mark.
07:28But as announced, thanks to the Pythagorean Equality, we will be able to calculate the length b, c.
07:35And yes, we are told that b, c squared equals 117.
07:39That is to say that b, c, x, b, c equals 117.
07:44In fact, I am looking for a number whose square is 117.
07:48Now, this may seem complicated.
07:53So I would like to find something that multiplied by itself gives me 117.
07:58The answer lies in the square root.
08:01The square root is the inverse of the square.
08:04I invite you to watch this video if you want to learn a little more about the square root,
08:09how it is defined and how it works.
08:11Well, in any case, what we are going to do is use the calculator because the calculator gives an approximate value of the square root.
08:20And yes, very often, the square root does not give us an exact value.
08:24Well, we are going to use it and ask the calculator to tell us how much b, c is.
08:30b, c, let's repeat, it's the square root because 117.
08:33So, I forgot to specify square root squared, it's written like this.
08:37A v that covers the entire number.
08:41Well, it's not exactly a v historically, it was obviously an r as the root.
08:47So, we see that the calculator gives us an approximate value.
08:51We're going to keep a value to the 10th and we're going to take 10.8.
08:54So, which means that if I replaced 10.8 squared here, it should give me something quite close to 117.
09:10We can test it and we can see that by entering 10.8 squared, we are not very far from 117.
09:15Of course, we do not end up exactly at 117.
09:20To do this, we would have had to be able to write all the decimals of square root squared 217.
09:25But, ah, there are an infinite number of decimals.
09:28So in fact, square root squared 217 does not have a decimal notation.
09:32But I repeat, I also explain it in detail in the other video that deals with the square root.
09:38Ah there.
09:38So we see, well here again, I repeat it, ah also without detailing the wording, but we see that the Pythagorean theorem allowed us to obtain an approximate value of the length of the side that we were missing in our right triangle.
09:53Another application of the Pythagorean equality.
09:57Well, the Pythagorean equality will allow us to check whether a triangle is right angled or not.
10:02To do this, we will use another theorem that also follows from the Pythagorean equality and is called the Converse of the Pythagorean Theorem.
10:13It's a bit long to write but it's quite important.
10:16What is a converse?
10:18Well, I made a video about it that I invite you to watch if you want to know a bit more, but I will explain it very briefly and very simply.
10:26In a theorem, there is always or most of the time a condition and a conclusion.
10:30If something, then in conclusion, we have something else.
10:37Well, the converse of a theorem is simply when we take the condition and the conclusion and we exchange them.
10:44Earlier in the theorem, we said that if a triangle is right angled, then I have the Pythagorean equality with the square and so on.
10:52Well, here, in the converse, we see it here, it's the opposite.
10:55If in a triangle, I have the equality, then conclusion, the triangle is right angled.
11:02Which means that here, unlike before, we don't know that the triangle is right angled.
11:07And besides, that's what we want to prove.
11:10Let's look at this triangle.
11:11Triangle 5, 1, 2, 1, 3.
11:14I know the lengths of these three sides, but I didn't code that it is right angled because I don't know.
11:19I would like to prove that this triangle is right angled.
11:23And that's where I'm going to use the converse of the Pythagorean theorem.
11:27But be careful, to use the converse of the Pythagorean theorem, you have to know the three sides, the lengths of the three sides.
11:34If we only know two, then well, we may have to do something else with angles and possibly trigonometrics, but in any case, we will not be able to immediately apply the converse of the Pythagorean theorem.
11:46So here, I would like to prove that this triangle is a right triangle.
11:51And to do this, well, we are going to check if the Pythagorean equation works, that is to say if the long side squared is equal to the sum of the squares of the other two sides.
12:01So yes, I say the long side, I no longer say the hypotenuse.
12:07Simply because I do not know for the moment that this triangle is a right triangle.
12:12So I cannot talk about the hypotenuse for a triangle that is not a right triangle.
12:16But one thing is certain, is that if this triangle is a right triangle, then of course the longest side will be the hypotenuse.
12:23So in my calculation, in the calculation of the Pythagorean equality, the length that I'm going to put on its own will be the longest side.
12:33And so the sum of the squares of the other two will be the other two sides, which means that if there is a hypotenuse, this hypotenuse necessarily measures 13.
12:41So I'm going to start by calculating BC squared on its own.
12:46That is, 13 squared on its own.
12:49And then, I'm going to calculate the sum of the squares of the other two.
12:53So AB squared plus AC squared.
12:57So that's, that is, 12 squared plus 5 squared.
13:01And so if BC squared is equal to AB squared plus AC squared, we can say that our triangle is a right triangle.
13:11So let's calculate 13 squared.
13:14Okay, that's 169.
13:1712 squared, well that's 144 and 5 squared 25.
13:22I do 144 plus 25 and I get 169.
13:26So we have 169 on each side, which means that BCO squared is equal to AB squared plus AC squared.
13:36And so we can apply our reciprocal of the Pythagorean theorem to conclude that triangle ABC is a right triangle.
13:44We can even specify that's noted.
13:46But I repeat again, the wording here is not intended as an example.
13:50Don't hesitate to join the playlist with all the videos on Pythagoras to see in more detail how to write it.
13:58Last thing, to finish, if we hadn't found the same thing,
14:02well that would mean in that case that we couldn't apply the reciprocal of the Pythagorean theorem.
14:07And so that would mean that our triangle is not a right triangle.
