 Hello friends and welcome to another session on geometry and we have been discussing triangles. Today, we are going to take up another theorem which is more commonly known as the Pythagoras' theorem. Okay, so you have been studying about Pythagoras' theorem for a long, long time. So it is not new to you. So this is called Pythagoras' theorem. Now why it is called Pythagoras' theorem? The story goes that Pythagoras in around 6th century BC proved this theorem. It is not that this theorem has not been there before Pythagoras' era, it was there. In fact, if you look at Indian sources, Indian mathematical sources as well, so there is this scholar called Bodhayana, right, Bodhanyana and he dates around 8th century BC, right. So before Christ 8th century BC, Bodhayana actually also gave the proof of this theorem in the series of books called Sulbasutras, right. So the book he wrote, all these people wrote, called Sulbasutras, right. So these are post Vedic literature, okay. So in these books, the proof for these kind, this theorem has been discussed. Okay, so let us not get into the history part of it more. Let us try to find out the proof or let us say try, let us understand this theorem and then try to prove ourselves. Now what does this theorem say? It says in a right angle triangle, the square of the longest side which is also known as hypotenuse, you know this, hypotenuse. So the square of the hypotenuse is equal to the sum of squares of other two sides. So in this case, we can start with the formal proof starting with let us say what is given. So it is given that in triangle ABC, ABC, we have angle B is equal to 90 degrees, okay. Angle ABC is 90 degrees other. And we have to prove, what do we need to prove? We need to prove that the square on the longest side which is AC. So AC square is equal to sum of the squares of the other two sides. So that is AB square plus BC square, okay. And there are lots of proofs for this theorem, you know, more than 20-25 proofs are existing. So we are going to use the concepts of similar triangle to prove this theorem here, okay. So you would have studied similar triangles by now. So let us now try to prove it. So for that we need a construction. So the construction is I have to drop a perpendicular BD onto the hypotenuse that is AC, okay. Now let us try to see or observe some similar triangles. So how to go about the proof? Now this particular diagram is pretty common in similar triangles and you can see triangle ABC is definitely similar to triangle the upper one. So this one I can draw. So this one is similar to the main triangle and this will be ABB, is it it? So these are two similar triangles, why do I say that? Because angle A is common to both triangles. Look carefully, angle A is common and there is a 90 degree as well. So angle V here is 90 and angle D here is 90, is it it? So hence which criteria we are using guys? So the criteria A is AA, AA that is angle angle similarity criteria, similarity, similarity criteria. Is it it? By this criterion we can very well say that the two triangles are similar. Now the moment these two triangles are similar what do we conclude? We can conclude that the let us say side AB, the sides will be proportional, right? So AB upon AD is going to be equal to AC upon AB, okay. So these are the two ratios for the two triangles. How do I get that? So let us say AB, AD so I wrote it back here and then AC and AB are corresponding sides. So can I not rearrange them and write as AB square, AB square is equal to AC into AD, right? So there is this first relation and let us name it as 1, first relation is 1, okay. Now let us take a triangle ABC again but this time we will choose the other triangle that means this one. So let us choose this triangle and you can see this is similar to triangle, ABC will be similar to BDC, BDC. This is these are these two triangles are similar so angle C is equal to angle C. Why? Because angle C is common in both and angle B again 90 degree and in this case angle D 90 degree. So these are two, these are the two angles which are equal. So guys if that is so then again we can say in this case BC upon DC is equal to AC upon BD, right or BC rather. So BC upon BC is equal to AC upon BC and then rearranging again you can very easily say what is this? BC square is equal to AC into DC, right? So this is equation number 2, equation number 2. Now what we are going to do is we are going to simply add these two equation 1 and 2. So you can write from 1 and 2, right? From 1 and 2 what do we need to do? Just add so this is AB square plus BC square on the left hand side and on the right hand side you will get AC into AD, AD plus AC into DC, okay? So now we can take AC common. So let's take AC common and you will get here AD, AD plus DC. Now what is AD plus DC? Look at the diagram. If you look at the diagram AD plus DC is AC itself. So hence this can be written as AC, AC into AC, yep which is nothing but AC square, AC square. So you can see AB square plus BC square is coming out to be AC square and this is what we wanted to prove. See this was what we wanted to prove and hence proved so we can say hence proved, okay? So I hope you understood how to prove Pythagoras' theorem or you can also call it Bodhainas' theorem. So hence we proved this theorem. There are multiple other ways. You can use congruences, you can use area, multiple ways of proving the same theorem. So hence we have proved it using the concepts of similarity, right? Next video we are going to see the converse of this proof. That is if this relation holds that is the sum of one, the square of one side is equal to sum of the squares of the other two sides, then the triangle is bound to be a right angle triangle, okay? So let us prove this in the next session.