 Hello friends, welcome to another session and in this session. We are going to prove another theorem It says that the diagonals of a trapezium Divide each other proportionally. So let's start with a diagram first. So let me draw a trapezium so So let me draw a trapezium first. Okay, so this is a trapezium and It says that the diagonals of a trapezium divide each other proportionately. So let's also draw the diagonal. So let's say this is first diagonal Let me re-draw it. So this is This is the first diagonal All right, okay, and the other one is this Okay, good. So we draw the We have drawn the trapezium now Let me name it. So let's say this is a b c d a b c b Okay, so let's write the All right, what is given what is given guys? So a b c d is a trapezium a b c d is a trapezium. What is a trapezium a trapezium is a quadrilateral having Having one pair of sides as parallel, right? So let's say a b is parallel to c d in this case Okay, so a b is parallel to c d. So I write it here a b is parallel to c d Okay, now to prove To prove what do we need to prove we need to prove Let's say this is point of intersection o of diagonals a b a c and b d. We have to prove that a o by o c Is equal to b o by o d This is what we need to Okay, so let's see how to go about it. So again, you can see there are triangles and the ratios of sides are there So somewhat we were, you know We are we are getting a hint of using basic proportionality theorem So for basic proportionality theorem, I need to have a triangle where two parallel Sides are there or you know one side is parallel to one other side which divides the two other sides So what I'm going to do is I need to do some construction. So let me construct something so construction What is the construction? So I'm going to do this and I'm going to join Or let's say I'm going to I'm going to draw o e parallel to o e is parallel to A b now the moment I do this It always automatically becomes parallel to dc, isn't it? Now the moment o e is parallel to A b in the triangle you can see we can write few ratios now, right? So let's go back to or let's do the proof now So what is the proof in? Let's take triangle a dc first a dc. Can you see a dc? a dc right in triangle a dc what is given e o Or o e is parallel to dc Right construction by construction Right now therefore By bpt basic proportionality theorem You can say what can you say you can say a e upon E d is equal to a o upon o c Right let it be one Right and now I'm writing it here Now consider triangle the other triangle e o is parallel to a b as well So these are all parallel lines. So e o is parallel to a b Correct. So in triangle o In triangle a bd a bd d happens to be the vertex in this case So let's say then What can we say we can say since E o again is parallel to the base a b therefore a e by Or rather e d or d let me write d e first so that it becomes easier for you to relate to it So d e upon e a is equal to d o upon o b Isn't it and if you reciprocate it you will get e a upon d e e upon d e is equal to o b upon o d Or d o whichever way so this is two Now if you look carefully one and two the left hand side the left hand side of one and two are same right only the Style of writing or you know the order of points is different So you can rearrange it and say a e by e d Is equal to o b by o d same thing Correct. So now from one and two you can write from from one and two Very easily you can write a o by o c Is equal to o b by o d And this is what we need it to prove Isn't it so now you see the diagonals are bisecting or not bisecting sorry diagonals are dividing the Dividing each other in equal ratio that is they are proportionally dividing each other So that's what the theorem was all about so Diagonals of or trapezium divide each other Proportionally we prove that in fact the converse is also true for this so we'll see we'll see the converse now