 All right friends welcome again to another session on problem-solving related to triangles We are doing basic proportionality theorem and its converse and their application in different types of problem-solving Now in this question, it's been given that two triangles ABC and DBC Lie on the same same side of the base BC. So you can see the figure I have drawn the figure BC is the base and two triangles ABC ABC and and BCD right already BC lie on the same side Now from a point P on BC where you can see the point P is here from the point P on BC PQ is parallel to AB and PR is parallel to BD. Okay, so you can see PQ this one is parallel to PQ is parallel to AB so I can put these arrows here So PQ is parallel to AB and PR is parallel to BD. So PR is parallel to BD Okay, now they meet at AC in Q and DC in R which is shown in the figure. We have to prove that QR is Parallel to AD. Okay, so this is the question now Approach let's talk about the approach we will take So what should decide our approach now clearly there are triangles in world and there are parallel lines in world Okay, so that they're the moment there are triangles and parallel lines The first thing which comes to our mind is basic proportionality theorem or the Thales theorem, right? So somehow it is hinting upon that we are going to use BPT or basic proportionality theorem So we can actually do the reverse. So for example QR is to be proven to be parallel to AD Now by converse of BPT if somehow we prove that so this is you know, this is how you should be Approaching it. So now I'm doing some kind of Analysis here. So they're asking me or us to prove that QR is parallel to AD now what we'll do is let us say that QR is parallel to AD then what will happen Then clearly by BPT, we know that if this is true then CR upon RD must be equal to CQ upon QA and that means what if we somehow prove this This particular thing then we know we will lead to this Correct. So let us say if we can prove CR by RD is equal to CQ by QA which in a way Is there how let us see now Let me now keep it. So now you know the target. You have to prove that QR is parallel to AD Now there are rituals. What are the rituals or ritual is given? What is given? Q PQ is parallel to AB Okay, and and what is given PR is also parallel to PR is parallel to AD Okay, and what is to be proven? So to prove you write to prove to prove QR is parallel to AD Okay, and here goes the proof Okay, how so what you will do is we will write in triangle See observed triangle a b c. Okay in this triangle PQ is parallel to the third side AB right and we know that by BP team that case what will happen? We will write Cp by Pb is equal to CQ by Qa and within brackets you must write by Bpt or with Thales theorem Okay, now you see observe Observe here. We have got CQ by Qa, which is on the Russian side of the desired result Right, so somehow if we prove that CR by Rd is also equal to Cp by Pb here, then we are done Okay, let's see how now In triangle, so I'm writing here itself in triangle Now we are coming from here to here. Okay, so in triangle Cad Okay, we know that PR is parallel to AD I'm sorry, not Cad it is actually Cbd Okay, so not Cad we are going to deal with Cbd and in Cbd PR is parallel to Bd Okay, so again by Bpt we can write Cp upon Pb So this upon this should be equal to this upon that Right, so hence we can write that as CR by Rd Okay, so let us say this was equation number one and this was equation number two So hence guys from one and two From one and two What do we see we can equate the RHS Of one and two and we can write Cq upon qa is equal to CR upon Rd see this is the desired thing which we wanted here We got the same thing now this will imply this implies what? by converse of by converse of Bpt Rq is Parallel to AD Okay, Rq is parallel to AD and you can write in triangle which triangle? Cad D Right now, so if you see in this triangle C This one let me draw it from that, you know C and then Cad is this Right See Cad is Cad is this triangle and in that triangle this thing So basically this side by this side was equal to this side by that side Is it it? So hence We can say that These two lines are also Parallel okay, so by application of Bpt and by application of Converse of Bpt we could prove that the Given lines are parallel so what is the learning guys learning is we see a triangle and there are discussions on parallel lines you know what could be probably used and that's basic proportionality theorem and With some manipulation with the direct Bpt and its converse We can solve these kind of problems