 Hello friends Welcome again to this new session on Triangles now in the last session we saw a very important theorem and we also saw its proof and scored basic proportionality theorem Now here what we are going to do is we are going to study some of the corollaries to basic proportionality theorem So corollaries are nothing but something some results which are which can be directly seen from some established result So, you know, it's not new or a different theorem is the same Theorem but with different manifestations. So or the same theorem leads to similar type of results. So those are corollaries. So let us now You know, see what our corollaries corollaries can we see here now? Let's recap BPT first that is basic proportionality theorem. So by Basic proportionality theorem, which is also called as Thales theorem. We learned that right basic proportionality basic proportionality so Pardon my spelling proportion proportionality right So basic proportionality theorem or Our other name was Thales theorem Thales Theorem now, what does this theorem say? It says that in a triangle if DE one line intersects or Cuts two sides of the triangle such that D is parallel to BC It's parallel to one of the sides Then we know that AD upon DB is equal to AE upon EC Correct. This is our Thales theorem. Now, let's see some corollaries to it. So corollary one So if I add one to both sides to the Equation add one by EC plus one is Equal to if you see what will it be? It will be AD take LCM and do the basic mathematics divided by DB is equal to AE plus EC divided by EC Now if you see what is AD plus DB guys AD plus DB is simply AB, isn't it? So hence we can write AB Upon DB is equal to AE Sorry AC AE plus FC AE plus EC on the right hand side is nothing but AC AC upon EC, isn't it? So hence this is the first corollary So first corollary is or the first result of BPT was that this Upon this was equal to this upon this, isn't it? Now the first corollary is full thing divided by this part is Equal to again on the other side full thing divided by this part. Now if you see this this has resulted from a rule in proportions called component dough Component dough. So from there also we could have said it directly component dough is nothing but if A by B is equal to C by D Let us say there is a there are two ratios which are equal and come rule of component This is A plus B upon B is Equal to C plus D upon D for example 1 by 2 is equal to 3 by 6, isn't it? So according to the rule of component dough, you will get 1 plus 2 by 2 is equal to 3 plus 6 by 6 Which is clearly if you see this is 3 upon 2 is equal to 9 upon 6 both are equal to 3 by 2, isn't it? So this is called component dough rule. So from component dough rule We had one ratio and we converted this ratio into This right. So this is also Valid or you know by BPT we can say this also can be directly said isn't it now Corollary 2 would be nothing but You invert this ratio the final ratio what you've got you invert it inverted means take the receivable. So hence, you know DB upon AB will be equal to EC upon AC, isn't it? This is this rule is called invert and dough Invert and dough means what if 2 by 3? So I'm writing the example of invert into here 2 by 3 is equal to let's say 4 upon 6 Then if you invert or take the receivable, then also it will be same Okay, so this is another corollary another corollary could be nothing but If you see Another corollary could be let us say, you know, let's start from The first one itself. So let us say I start from here. So AB by DB Okay, AB by DB is equal to AC by EC, isn't it? Now, what can I say about DB? I can say AB upon DB if you see what is DB here DB here is nothing but full minus AD, right? AB minus AD so I can write AB as a DB as AB minus AD AD just check once again What is we're talking about? We're talking about DB. That means we are talking about this length DB Now DB is nothing but full AB minus AD So AB minus AD I have written here. Now next AC upon EC. Now, what is EC? Let's see what is EC EC again is nothing but AC minus AE AC minus AE, so let us write AC minus AE right and then Use the invert into again. So you will get what you take the receipt portal By AB. In fact, you could have started from second corollary directly, but never mind. So this is AC minus AE by AC Right, this can be further written as AB upon AB. Now I'm splitting the fraction minus AD upon AB is equal to AC by AC minus AE by AC Now if you see what is AB by AD, it is nothing but one minus AD by AB is equal to one minus AE by AC So if you simplify this this one and this one will go this minus sign and this minus sign will also go So hence it becomes AD by AB is equal to AE by EC. This is another corollary. So what all corollaries did we learn? So I will just summarize here first by BPT you had got AD by DB is equal to AE by EC, is it it? This was the basic one and the first corollary to it was nothing but AB by DB is equal to AE by EC By adding one you had got that. See we had got this one Right, AB by DB is equal to AC by EC here Right, now we this was from component, isn't it? The rule of Component related to proportions. Now we did invert endo and we got DB by AB is equal to EC by AC and this is called invert endo invert Endo right and third one with some mathematical manipulation we got What do we got as a third one? AD by AB So I'll write here AD AD by AB is equal to AE by AC Right, and then you can you can actually derive many more. Let's say now I will use something called alter nendo. What is that? Alter nendo, and what is alter nendo? Alter nendo says that if 1 by 2 is equal to 3 by 6 then 1 by 3 is equal to 2 by 6 So if you see I just interchange these places these numbers right so hence Using alter nendo in all we can actually have many more corollary. So if you see AD by AE if I take the first one and Apply alter nendo that means you exchange these values So what will you get? AD by AE will be equal to DB by EC then here in corollary 1 if you Apply alter nendo, you will get AB by AC is equal to DB by DB by EC Right? No, likewise you can all all other or you know If you apply let us say on the second corollary you will get DB by EC is equal to AB by AC which is same as the fifth one and then Yeah, and and if you apply the same thing on the third one you'll get AD upon AE is equal to AB Upon AC Another way to look at it. So what did I do? I just applied alter nendo on this and I got this Right, and then there are multiple others. What others like you know You can do cross-multiply on on the first one here are the basic BP T and what will you get? You will get a D into EC Is equal to AE into DB? Isn't it just cross multiplication? Similarly for all others you can get a Corresponding corollary. So for this you will get AB into EC Is equal to DB Into AC. Okay. Similarly here. What will you get? You'll get DB into AC is equal to AB into EC which is same as the previous one So I'm not going it going through this. So if let's say you do this this will be AD into AC AB into AC is equal to AB into AE and likewise you can you can have so many corollaries, isn't it? Why am I Going through these many Corollaries is simple that while problem-solving there could be cases where they're asking you to prove something of this sort Yeah, so whenever you see something of this sort, you must know that This is this must be or this might be a case of application of basic proportionality theorem, right? So you can see so many corollaries could be drawn using just one Okay, so please keep these in mind and whenever such problems are there Where you are encountering such kind of expressions to be proved then? You can try basic proportionality theorem over there