Understanding Thales Theorem and Its Applications

hi everyone welcome to this class where we'll be looking at the tales theorem it's also known as the basic proportionality theorem so let's understand what is this theorem we'll also look at its Converse and some important questions and I'm going to make these math concepts absolutely crystal clear for you so let's go ahead and get started before we begin I just want to say do check out the other courses on our website you know we have physics chemistry biology and Maths for cbsc Class 8 9 and 10 so guys if you haven't taken the other courses do take them and do share it out with your friends so what is Tales theorem please remember Tales theorem is also known as basic proportionality theorem and if you want to understand in a simple way what does this theorem say that if you have a triangle okay and you draw a line parallel to the one of the sides of the triangle so let's say we draw a line D parallel to this side now it need not always be the base here let's in this case it's looking like the base it can be to any side of the triangle so here de is drawn parallel to this side BC then this theorem says that it will intersect the triangle at these two distinct points this line which is parallel at d and e in such a way that these two sides the ad and AC side is div divided by this parallel line in the same ratio so please read the theorem and let's understand it if a line is drawn parallel so line basically means here the line de if a line so for example here the line de is drawn parallel to one of the sides BC so clearly you can see D is parall to the side BC and it is intersecting the other two sides at distinct point d and e so can you guys see the distinct points here D and E the distinct points here are d and e then the other two sides other two sides means not the side which is parallel we are not talking about BC the other two sides ab and AC so what sides are we talking about ab and AC are divided by this parallel line in the same ratio so what is the ratio part we interested in that means this a is to DB the ratio of ad is to this length DB is going to be equal to AE is to EC do you guys get the idea what is the ratio here so basically this theorem is saying if D is parallel to BC then a d divided by DB is going to be equal to AE ided by EC this is basically Tales theorem or also known as basic proportionality theorem so you can just remembered with this simple diagram if you draw this parallel line you can see that this ratio of a by DB is going to be equal to AE by EC and ratio means the lengths of these sides so let's say if this is uh so if you take an example here let's say this length is 2 cm and this length is 4 CM this length is let's say 3 cm and this length is 6 cm now when you've drawn this line de parall to BC clearly you can see what is the ratio a is to DB so in our example a is to DB is 2x 4 which is half here and similarly AE is to EC is 3x 6 which is again half so can you guys see that it is being divided in the same ratio this is exactly what is Tales theorem or basic proportionality theorem what is the proof of tales theorem so if you guys look at it intuitively it seems to make sense and we have done similar triangles also so can you guys see that these two triangles are going to be similar to each other how if you take a common angle and then you know these angles are going to be corresponding angles even these are corresponding angles so by a A or AA a we know that a is going to be similar to ABC so you can imagine that yeah things are going to be in proportion but here we are not comparing ad is to the entire AB we are saying ad is to DB so it's a little different from similarity because we are saying a is to DB is equal to AE is to EC these parts that the parallel line is dividing it is dividing this side AB into these two parts these two parts are in the same ratio as these two parts clear but let's not worry about similarity right now but intuitively you can get that this seems to make sense but the proof doesn't use similarity there's a very interesting proof based on area of a triangle which you're going to look at that how will you prove Tales theorem using area of triangle concept let's take a look so proof may look complicated when you see it first time but if you go through it carefully and I'm going to explain the trick to you I think then you'll find it better very simple so again see this is the theorem we are trying to prove so we have written the theorem and again you start off with what is given given is the triangle ABC and you can take any uh you know any line parallel to one of the sides we have taken de parallel to BC so you guys can see we have taken D parallel to BC here and D is intersecting AB at D and AC at this e so these were the two distinct points and obviously what do we need to prove prove that ad by DB so this is Tales theorem a by DB is equal to AE by EC this is what we need to prove here now how will you prove this since we are going to use area of triangles you know area of triangle what is the formula do you guys remember area of triangle is half Base multipli by the height and height in a triangle always means perpendicular height you can't have a slanting height right so area of triangle always involves half base times perpendicular height that means since we are using area of triangle we're going to be using area of triangle concept we need to draw some perpendiculars now we we are interested in the area of this triangle let's say a so we will drop these perpendiculars from D to this line AE so we have dropped a perpendicular here am and then we have constructed another perpendicular so this is very important step where you need to do these constructions so draw these two perpendiculars DM and en n which are perpendicular to AC and ab respectively and also join DC and be so you can see the constructions are done here okay so it's looking complicated that hey we have added so many things here but trust me it's going to be really simple okay so now what are we going to do we are going to start off with proving uh with using the area of this top triangle a d e so I want everybody to focus on this triangle on top triangle a d e what is area of triangle formula you guys know it is half base into height now if you're looking at triangle a what will be the area half base into height so if I consider this as my base if ad is my base what is the height here clearly the perpendicular height is en n do you guys agree this is going to be my H so I can say area of triangle ad is half * a * the perpendicular height n do you guys agree please see if this is clear to you area of triangle a is half * base multiplied by the height so this half base into height everybody is clear half base into height is the area of the top triangle now since I'm interested in the ratio of a is to DB now I'm going to shift my attention to the triangle below which triangle I'm talking about triangle dbe so we are going to look at the area of this triangle triangle d b e this time this guy so I'm looking at this triangle what will be the area of triangle dbe so if you guys carefully take a look so for this triangle let's say DB is my base if I make DB as the base this length is not the same as this I'm just calling it B for base what is the perpendicular height here please remember that perpendicular height need not be inside the triangle it could lie outside so do you guys agree if you look at this triangle dbe e triangle DB e if DB is the base then the perpendicular height is it is that same guy which is outside the triangle so please visualize it like this so if this is my triangle dbe the perpendicular height can be outside this is the height en n okay so again I'm going to substitute the area of triangle DB is going to be the base DB multiplied by the same perpendicular height en n so if I take a ratio of area of these two triangles the top triangle a d e divided by the area of triangle dbe e what am I going to end up with I'm going to get I basically need to divide these two this divided by this so what's going to happen will get cancelled if I do a division this guy is going to get cancelled so what will I get a by DB and half will of course get canceled clear so if you're finding a ratio of the area of these two triangles ad and DB you can see half will get canceled n will get canceled so I'm going to be left with ad / by DB and AD by DB is very important because this is what we looking for in the's theorem we have to prove a by DB is equal to e AE by EC we have to prove that they are in the same ratio so the main thing what we are doing here we are looking for a ratio of area of triangle a e divided by area of triangle DBC sorry dbe these two triangles so a d the top triang triangle and this triangle if you do the ratio of their areas you will find half will get cancelled the perpendicular height is same because the E is the vertex so you're dropping the same perpendicular so what you're left with is AD / DB that's all you're left with so that's great now let's take a look if we do the same thing but now on this side so instead of considering that is the base if we again look at our top triangle a a d e and this time consider a e as the base so what will my perpendicular height be this time it's going to be DM so if I consider area of triangle a then with AE as the base the perpendicular height is going to be DM this will be my height H so what will be the area of the triangle ad half time AE uh this should not be EC oh sorry half time a e * DM so you can see area of triangle is going to be half * a e * the perpendicular height DM and what is the area of the triangle below that that is the triangle d e c or ecd whatever you want to call it area of triangle ecd this time this will be the base and the perpendicular height is going to remain the same if you're looking at triangle a d c because this is the base and this is the perpendicular height same so triangle of uh area of triangle ecd is going to be half times the same perpendicular height DM but this time the base is EC for the lower triangle so again if you divide what's going to happen half will get canceled if you find the ratio of area of these two triangles half will get canceled DM will get cancelled so we are left with AE / EC and that's exactly what we want for the theorem AE / EC so now if we compare so here what we have got ratio of area of the two triangles ad by DB here we have got ratio of area of two triangles a by AC we need to prove that they are equal now for them to be equal this area guys need to be equal so what you can see here here we have the same in the numerator we have the same triangle triangle a and in this second equation also we have the same triangle ad but the denominators are different here we have triangle dbe which means we looked at this triangle the one on the left and here we have the triangle ecd the one on the right but if you guys carefully look at these two triangles dbe e and D EC or what did we call it ecd can you guys tell me what is the area of these two triangles are they equal or not so again I'm saying triangle dbe and triangle e CD do they have same area the answer is yes why look carefully if you consider now Ed as the base if you consider the common base you can see that both triangles have the same perpendicular height if you consider this as the base both the triangles have the same perpendicular height because they are between parallel lines so clearly we can see that these two triangles between the parallel lines and formed by these dotted lines are having equal area so therefore what can we say that triangle dbe and ecd are on the same base and between parallel lines so their area is going to be same dbe and ecd have the same area so the denominators are same so that means area of ad by dbe is going to be same as area of ad by ecd since denominators are same therefore we can clearly get a d by DB is equal to AE by EC and so we have proved the tales theorem Tales theorem could also be proved using similar triangles but we did not use that approach we used the area approach it looked a little complicated because we were doing ratio of areas and we were considering a lot of triangles here but in the end you saw that it all worked out same because these two triangles at the bottom have the same area so this one and this guy here both these triangles ended up having same area so a d by DB is equal to AE by e easy we have proved the tales theorem or the basic proportionality theorem so please practice this proof first time you see it it looks complicated but it is all working with the area half base into height the area formula and then considering the correct base and the correct perpendicular that's it then just writing the ratios you will get the answer now what are some interesting corollaries from the tales theorem so one important coroller is that in if you have a triangle ADB like this where you have de parallel to BC and it intersects you know the other two sides at these distinct points a at d and e then what we can say by theales theorem we saw that a d by DB is equal to AE by EC right we talked about that this a by DB so this parallel line is dividing the two sides in the same ratio is equal to AE by EC but not only that also a d also AB so either you take these two parts a d by DB is equal to AE by e c or you can take the whole line AB / DB is going to be equal to AC the entire length divided by EC okay and similarly you can take AB this entire length AB ided by ad so this time you take only the top part is going to be equal to a divided by AE that means not only you just take a ratio of these two parts you can even take the full length and one part they will all be in the same ratio so all these ratios are going to be equal and you guys can check that with an example so for example if you take like we had taken uh let's say this one was 2 cm we had taken this as 4 cm this one as 3 cm and 6 cm so let's confirm so if you confirm the first one a by DB so what is ad by DB we saw that is 2x 4 will be equal to 3x 6 AE by E because we know half is equal to half so this ratio is correct now if you want to prove this one AB by DB what we get here AB is this entire length that means you add it up 2 + 4 so I'm getting 6 divided by DB 4 needs to be equal to AC the entire length is 6 + 3 9 divided by EC which is 6 and now you guys can confirm and check this is going to be 3x2 this is also going to be 3x2 there you go and you can check the last one also so basically you can take either just part of of the side or the entire side but again in the same corresponding way right so here if you take the full length AB it's going to be 2 + 4 6 divided by just the top part now two and this side is going to be AC the entire length 6 + 3 9 divided by just the top part AE which is three and if you guys check both the 6X 2 is 3 9 by 3 is 3 so you can check with some simple example you can see all these ratios whether you take the part of it all the entire thing they're all going to be equal so here remember you can remember this with this diagram that if you take this entire length divided by just this part and this entire length divided by just the top part they are going to be in the same ratio so how do you prove this coroller it's very easy because by theal theorem we know a by DB is equal to a by e now what is the trick is you add one to each side so the trick trick of proving the cor corer is adding one to the left and right hand side when you do that what happens the denominator is going to be DB so the numerator is going to become ad plus DB because the LCM is DB so ad plus DB is what if you guys take a look a plus DB is nothing but the entire side AB so this whole thing if you add it up ad plus DB is nothing but ab and similarly AE plus e c is nothing but AC so you can see so very simple you start off with Tales theorem just add one on both sides and you can easily prove the coroller similarly you can prove the second coroller also you start now not with ad by DB you start with the reciprocals TB by a is equal to EC by AE so you take the reciprocal this time now again you add one and now when you calculate you're going to get the same thing this entire length AB divided by ad the top part is going to be AC / AE so very simple to prove it you just add one on the left hand side right hand side in one case you start with the tales theorem in another one you start with the reciprocal of the tales theorem done how are we getting AB by a so see we started with DB by a and when you do+ one the LCM is AD so this will go DB + a or you can think 1 is ad by a so the numerator is DB plus a now look at the diagram what is DB + a this entire length AB so I'm replacing this DB + a d with AB similarly I'm replacing AC + AE with AC just simply adding it up now let's talk about the converse of tales theorem what is the meaning of Converse Converse means the opposite it's like sort of reversing the theorem so what do we say here if if a line divides any two sides of the triangle in the same ratio that means this time what is given to you let's say you have a line de and it is dividing these two sides in the same ratio so let's say this time uh this length is uh let's say 4 cm and this is 2 cm and here we have something like 6 cm and 3 cm so can you see the line de is divided both these sides in the same ratio because what is a by a / DB is 4x2 which is 2 and similarly AE / EC is nothing but 6x3 which is again two so you can see both are being divided in the same ratio which is 2 is to 1 if this happens then this means that the line that you have drawn de is parallel to BC so if this happens then we know that de is parallel to BC in tal theorem what was it it was opposite it was given that de is parallel to BC and we said that the ratios of these two sides are equal here it is opposite the ratios are given equal the that means based on it the line de must be parallel to the line to this side BC so is the converse or the opposite clear again how do you prove the converse of thales theorem so proof is very interesting let's take a look so first we'll write what is the Converse that we have to prove again we'll start off with the given triangle and what is given to us this is given that ad by DB is equal to AE by e c what do we need to prove that de this guy de is parallel to BC okay we will try to prove this in a very interesting way by the technique known as contradiction contradiction means what we will assume that it's not true and we'll prove that's not possible so it's like we're contradicting ourselves it's like saying you know uh like today is a Wednesday no no today is a Thursday no it's not a Thursday it's a Wednesday sort of like contradicting yourself so let's see what do we do here so what we going to do is we're going to say let de is not parall to BC so we want to prove D is parallel to BC but we say no no D is not parallel to BC so this is called the contradiction technique so what we want to prove we assuming it's false so let de is not parallel to BC so if this is not parallel we we going to say no we are going to draw a line which is parallel so we have drawn this random line DF and we are going to say let DF be parallel to BC I know it looks crazy DF doesn't look parallel to to BC but we're going to take it right so this is a crazy technique where we say we draw another line which is parallel to BC now if DF is parallel to BC this side BC that means I can apply Tales theorem in it what did Tales theorem say that the ratio of these two sides so a d by DB will equal to AF by FC I can apply th theorem because I drew this magical parallel line so it's like a Harry Potter line right we did this by Magic we have drawn another line and we are saying by Tales theorem this ratio is true but what is given to us it is given that a by DB is equal to a by EC this was what is given so if you compare these guys the left hand side is the same ad by DB is AF by AC uh AF by FC here ad by DB is a by EC so since the left hand side is same I can therefore say AF by FC is equal to a by EC the right hand sides I'm comparing that means these guys are equal AF by FC is equal to AE by EC clear what we have done till now where we drew a line which was parallel to BC so we'll start with this AF by FC is equal to a by EC and we'll do the same trick of adding one on both sides now when you add one something interesting happens the numerator becomes AF by plus FC what is AF Plus FC it is this entire side AC similarly here you get a plus EC so AE plus EC so again you get the entire side AC and now what's going to happen this AC AC will cancel AC means this side AC not air conditioner this entire side AC and it's going to get cancelled so what are we left with FC is equal to EC now carefully look at our diagram we are basically saying that this guy FC is equal to EC if FC is equal to AC this can only be possible if E and F are same yes because how can f c equal to EC it is only possible when E and F coincide only then this can be true if they coincide remember we said we had drawn DF parall to BC now if these guys are the same point F and E that means since you drew DF parallel to BC that means de is also parallel to BC and there we have proved the converse of Tes theorem that if the ratios are given equal that means the line is basically de is basically parallel to BC so it is not DF it is basically since they are the same line that means de and BC are parallel to each other so there we have done a lot of theorems the theorem the converse of tales theorem corollaries now let's get into some exciting sums here's the first question in the figure de is given parallel to BC so whenever you see this kind of triangle where this uh you know a line in the middle is parallel to this side you know that this is probably going to use Tales theorem or basic proportionality and now what is given to you that a is given as 4x - 3 a is 8x - 7 so all these lengths are given you need to find the value of x so how will you guys do it basically you will apply thales theorem so we are going to say Tales theorem or by basic proportionality theorem by Tales theorem since de is parallel to BC based on this we know that a d D / DB is equal to AE by e c yes it will this parallel line will divide the sides into equal ratios right ad by DB equal to a by e c now just go ahead and substitute so what will we substitute here ad is given as 4x - 3 / 3x - 1 and that's equal to 8x - 7 / 5x - 3 now how do you solve this equations because you have to solve for x so one simple trick is to cross multiply since there's nothing much simplification we can do so we can cross multiply here so basically we'll get 4x - 3 * 5x - 3 = 8x - 7 * 3x - 1 now I guess to solve this you guys will have to expand out the left hand side right hand side and see what equation you get and please solve that equation so can you guys do this for me so please multiply out these numbers and this one and see what you get so if you guys simplify you'll get something like 20 X2 - 12x - 15x + 9 right so you'll get all these numbers here and if you further simplify you're going to get 4x^2 - 2x - 2 which you can divide by 2 so you're going to get 2x2 - x - 1 = 0 what kind of equation is this quadratic equation because it has x² so you can solve it by middle term breaking Shar achara formula right so always first try middle term breaking and see if you can easily solve it and you look at the factors you know uh 2 and minus1 here so if you break it into - 2x + x you'll easily get these factors of x - 1 * 2x + 1 = 0 so either X - 1 is 0 or 2x + 1 is 0 which basically means either X is = 1 or x = -2 so let me write this again since it's not clearly visible so either X is = 1 or 2x + 1 is 0 which means X is - half now why do we say Min - half is not possible can you have two answers for X but Min - half is not possible why- half is not possible because because if you go and substitute min-2 what will you get 4 * -2 is -2 -3 so we getting a length of - 5 that doesn't make sense and same for the other cases so this is not possible because this will give us negative lengths so the only answer possible therefore X is equal to 1 is the final answer so that's the only answer which is possible here and you guys can substitute back and check 4 4 - 3 uh so if x is 1 4 - 3 that means this is 1 this is 3 - 1 this is 2 8 - uh 7 this is going to be 1 5 - so 2 so there you can see the ratios are same 1 is 2 1 is 2 that means we have found the correct answer and negative is not possible because it's going to give you negative lengths of these sides ad DB and so on so always whenever you get a quadratic you get multiple answers please check if your answer makes sense here's the next question p and Q are points on the sides de and DF of uh DF right this triangle such that DP is 5 cm so let's write down the values here DP is given to be 5 cm DQ is given as 6 cm de is 15 so this entire length de and qf this length is 18 cm so I know this diagram is not drawn to scale it's a rough diagram because qf doesn't look 18 right uh and this D is uh 15 this entire length so the question is is PQ parallel to EF what do you guys think so one is you could use this you know small side is to the big side but uh Tales theorem might be simpler because we want to prove PQ is parallel to EF if you just want to you know calculate this is to this that you can easily find out because what is the length of P 15 - 5 so we can do that the length of p is going to be D minus DP which is nothing but 15 - 5 so that's 10 cm so we know that our length P here is basically 10 cm so now to prove that if PQ is parallel to EF that means we are using Converse of thees right that if the sides are in equal ratio then it's going to be parall so let's find the ratio what is my ratio here DP is to PE so DP by pe is 5 by 10 which is basically half and what is my other ratio here DQ is to qf the other ratio is DQ ided by qf is going to be 6 by8 which is nothing but 1/3 so can you guys see the ratios are not equal so we can see therefore TP is to PE is not equal to DQ is to qf so I cannot apply Converse of thals for Converse of theales these ratios needed to be equal then we could confidently say that DQ is parallel to EF so therefore the answer is going to be therefore uh PQ is not parallel to the side EF and this is the final answer so see that's how we do it we find the ratios here and you saw the ratios of 1/3 and half so we can clearly say that they are going to be not parallel by Converse of thales theorem let's look at the next question this is very interesting where we have ABCD is a trapezium with AB parallel to DC and E and F are points on non-p parallel sides a and BC such that EF is parallel to AB so clearly you can see here AB is given parallel to DC in this trapezium and it says that EF is parallel to AB now we know if EF is parallel to AB that means EF will also be parallel to the other uh other side DC what do we need to show that AE is to e is equal to BF is to FC so since you have all these guys parallel can we simply say that this is true by theales theorem is this going to be correct because can I simply say that the ratio of these parts are going to be equal to the ratio of these two parts by Tales theorem am I done yes or no the answer is this is wrong why because reminding you that Tales theorem is for triangles there is no triangle here you are looking at a trapezium so this is not correct Tales theorem cannot be applied to trapeziums so what do we do very simple how about we break this trapezium or this quadrilateral into triangles because Tales theor we can apply for triangles so what you guys can do here go ahead and do a construction let's say join AC so if you join the diagonals of the trapezium so any one of the diagonal you're basically breaking it into two parts so there you can see so what you'll be doing construction where basically you join the DI diagonal AC now you can see you have nice two triangles and here we know that a Ab is parallel to EF and EF is also parallel to DC so you can write those points they're all parallel now if you look things become easy because the ratio of AE is to Ed so let's say this meets diagonal meets this guy at what should we call it let's say this is uh EF we can call this G so that means AE is to Ed D is going to be equal to AG is to GC by the's theorem because you can see these two parallel lines so now we can say by theales theorem if you're looking at triangle a DC so in Triangle ADC by theales theorem what do we know AE is to Ed AE by Ed is going to be equal to AG by GC so this we know the ratios of this is to this is going to be equal to this is to this now you focus on the other triangle triangle ABC in triangle ABC again this is parallel to this so that you should definitely write that all these three sides are parallel so now looking at triangle ABC again by thales theorem so what do we have we can say AG is to GC is equal to BF is to FC there you go so again we can say AG so this side AG is to GC is equal to BF is to FC because you don't always have to start from the top right you can take this is to this equals this is to this so there you go now you just equate these two so we're going to equate 1 and two so from 1 and two we can clearly see AG is to GC is common part so a by Ed will equal to PF by FC so we can say from 1 and 2 a e by e d is going to be equal to BF by FC there you go we are done proved so simple so the whole trick was breaking it into two triangles because you can't apply thales theorem to a quadrilateral like a trapezium so there you go okay here we have labeled this point as M or whatever you want to call it and there you can see the same same proof using the theorem you could also have used the diagonal DB you'll get the same thing so there you can see using both the things uh theil theorem on both the triangles so this is your triangle one and your triangle two you will get the same answer let's try another interesting question in the figure de is parallel to BC ad is given 3 cm so again in Geometry make sure you're marking everything here so a is 3 cm AE is 2 cm so this length is 2 cm BD is 2 cm so BD is 2 cm de is 6 cm and what do we need to find the length of BC this is the unknown so how do you guys plan to do this and you have been given that de is parallel to BC that we have marked in the picture here so can we find this side by applying Tales theorem can we say you know this by this equals this by this or how do we do it can we apply theales theorem for this one so Thal theorem according to thales theorem we know that if this de is parallel to BC so by theales theorem we know that when you have de parallel to BC then a is to DB the ratio of a by DB will equal to AE by EC because this parallel line divides these Parts into equal ratio but the problem is we don't want to find the length AC we want to find the length BC so can we apply Tales theorem here the answer is no so here basically Tales theorem or basic proportionality theorem does not help so what will you do here so because to find thales theorem doesn't help you find the length of the parallel side it talks about these the nonp parallel side right right the parallel line dividing the other two sides so how do you solve this so one simple trick you can do here when you can't apply Tales theorem again look back at the triangles because now you're looking at What triangle a and you might want to look at triangle ABC because you want to find the length BC what do we know about these triangles since the lines are parallel we know that this angle is equal to this angle by corresponding ing angles again you have parallel lines these angles are equal and of course this angle is common angle a so we know that these two triangles the top triangle ad and the bottom triangle ABC are similar so when you can't apply THS theorem go to the similar triangles so we can say that triangle a d e is similar to triangle ABC by a a or AA a angle angle criteria so now once they are similar you know you can find the ratio of the corresponding sides are equal that will help you find the length of BC so let's go ahead and do that so since they are similar to each other I can clearly say that a so the length ad is to the whole uh the other bigger triangle ad by AB is equal to D / BC corresponding parts of similar triangles are in the same ratio right these corresponding sides they are in the same ratio so this is what we do now you just go ahead and substitute the values here very simple ad is 3 cm AB is given to you the whole length AB will be 3 + 2 so carefully add it up this is going to be 3 + 2 which is 5 the length d uh the length D is given to you 6 cm so we'll subtitute that and we need to find the length BC so please go ahead and solve this so basically we are getting BC is 6 * 5 / 3 or BC = 6 * 5 / 3 which is nothing but 10 cm that's the final answer here so there you can see that using this similar triangle so what did we do so since Tales theorem did not work using similarity so similar triangles came to the rescue here since we wanted to find this side BC and you can see you'll get the answer very easily and so see by the a criteria you must show that the triangles are similar and we got the same answer 10 cm so hope the tales theorem or the basic proportionality theorem and its Converse is crystal clear to you now and we practice some very very interesting and important questions and do check out the other courses on our website we have physics chemistry biology and Maths for cbsc Class 8 9 and 10 so guys if you haven't taken the other subjects do take them and do share it out with your friends for ICC students once again we have physics chemistry biology and Maths for classes 8 9 and 10 and if you want to learn coding we have Python Programming we have Java coding both are great languages to learn computer programming and we have physics and chemistry for the Cambridge IGCSE which is the international board so please do share out our courses with your friends make sure you have subscribed to our YouTube channel we also have a Hindi channel it's called Manoa Academy Hindi to stay connected with Manoa Academy and keep learning

hi everyone welcome to this class where we&#39;ll be looking at the tales theorem it&#39;s also known as the basic proportionality theorem so let&#39;s understand what is this theorem we&#39;ll also look at its Converse and some important questions and I&#39;m going to make these math concepts absolutely crystal clear for you so let&#39;s go ahead and get started before we begin I just want to say do check out the other courses on our website you know we have physics chemistry biology and Maths for cbsc Class 8 9 and 10 so guys if you haven&#39;t taken the other courses do take them and do share it out with your friends so what is Tales theorem please remember Tales theorem is also known as basic proportionality theorem and if you want to understand in a simple way what does this theorem say that if you have a triangle okay and you draw a line parallel to the one of the sides of the triangle so let&#39;s say we draw a line D parallel to this side now it need not always be the base here let&#39;s in this case it&#39;s looking like the base it can be to any side of the triangle so here de is drawn parallel to this side BC then this theorem says that it will intersect the triangle at these two distinct points this line which is parallel at d and e in such a way that these two sides the ad and AC side is div divided by this parallel line in the same ratio so please read the theorem and let&#39;s understand it if a line is drawn parallel so line basically means here the line de if a line so for example here the line de is drawn parallel to one of the sides BC so clearly you can see D is parall to the side BC and it is intersecting the other two sides at distinct point d and e so can you guys see the distinct points here D and E the distinct points here are d and e then the other two sides other two sides means not the side which is parallel we are not talking about BC the other two sides ab and AC so what sides are we talking about ab and AC are divided by this parallel line in the same ratio so what is the ratio part we interested in that means this a is to DB the ratio of ad is to this length DB is going to be equal to AE is to EC do you guys get the idea what is the ratio here so basically this theorem is saying if D is parallel to BC then a d divided by DB is going to be equal to AE ided by EC this is basically Tales theorem or also known as basic proportionality theorem so you can just remembered with this simple diagram if you draw this parallel line you can see that this ratio of a by DB is going to be equal to AE by EC and ratio means the lengths of these sides so let&#39;s say if this is uh so if you take an example here let&#39;s say this length is 2 cm and this length is 4 CM this length is let&#39;s say 3 cm and this length is 6 cm now when you&#39;ve drawn this line de parall to BC clearly you can see what is the ratio a is to DB so in our example a is to DB is 2x 4 which is half here and similarly AE is to EC is 3x 6 which is again half so can you guys see that it is being divided in the same ratio this is exactly what is Tales theorem or basic proportionality theorem what is the proof of tales theorem so if you guys look at it intuitively it seems to make sense and we have done similar triangles also so can you guys see that these two triangles are going to be similar to each other how if you take a common angle and then you know these angles are going to be corresponding angles even these are corresponding angles so by a A or AA a we know that a is going to be similar to ABC so you can imagine that yeah things are going to be in proportion but here we are not comparing ad is to the entire AB we are saying ad is to DB so it&#39;s a little different from similarity because we are saying a is to DB is equal to AE is to EC these parts that the parallel line is dividing it is dividing this side AB into these two parts these two parts are in the same ratio as these two parts clear but let&#39;s not worry about similarity right now but intuitively you can get that this seems to make sense but the proof doesn&#39;t use similarity there&#39;s a very interesting proof based on area of a triangle which you&#39;re going to look at that how will you prove Tales theorem using area of triangle concept let&#39;s take a look so proof may look complicated when you see it first time but if you go through it carefully and I&#39;m going to explain the trick to you I think then you&#39;ll find it better very simple so again see this is the theorem we are trying to prove so we have written the theorem and again you start off with what is given given is the triangle ABC and you can take any uh you know any line parallel to one of the sides we have taken de parallel to BC so you guys can see we have taken D parallel to BC here and D is intersecting AB at D and AC at this e so these were the two distinct points and obviously what do we need to prove prove that ad by DB so this is Tales theorem a by DB is equal to AE by EC this is what we need to prove here now how will you prove this since we are going to use area of triangles you know area of triangle what is the formula do you guys remember area of triangle is half Base multipli by the height and height in a triangle always means perpendicular height you can&#39;t have a slanting height right so area of triangle always involves half base times perpendicular height that means since we are using area of triangle we&#39;re going to be using area of triangle concept we need to draw some perpendiculars now we we are interested in the area of this triangle let&#39;s say a so we will drop these perpendiculars from D to this line AE so we have dropped a perpendicular here am and then we have constructed another perpendicular so this is very important step where you need to do these constructions so draw these two perpendiculars DM and en n which are perpendicular to AC and ab respectively and also join DC and be so you can see the constructions are done here okay so it&#39;s looking complicated that hey we have added so many things here but trust me it&#39;s going to be really simple okay so now what are we going to do we are going to start off with proving uh with using the area of this top triangle a d e so I want everybody to focus on this triangle on top triangle a d e what is area of triangle formula you guys know it is half base into height now if you&#39;re looking at triangle a what will be the area half base into height so if I consider this as my base if ad is my base what is the height here clearly the perpendicular height is en n do you guys agree this is going to be my H so I can say area of triangle ad is half * a * the perpendicular height n do you guys agree please see if this is clear to you area of triangle a is half * base multiplied by the height so this half base into height everybody is clear half base into height is the area of the top triangle now since I&#39;m interested in the ratio of a is to DB now I&#39;m going to shift my attention to the triangle below which triangle I&#39;m talking about triangle dbe so we are going to look at the area of this triangle triangle d b e this time this guy so I&#39;m looking at this triangle what will be the area of triangle dbe so if you guys carefully take a look so for this triangle let&#39;s say DB is my base if I make DB as the base this length is not the same as this I&#39;m just calling it B for base what is the perpendicular height here please remember that perpendicular height need not be inside the triangle it could lie outside so do you guys agree if you look at this triangle dbe e triangle DB e if DB is the base then the perpendicular height is it is that same guy which is outside the triangle so please visualize it like this so if this is my triangle dbe the perpendicular height can be outside this is the height en n okay so again I&#39;m going to substitute the area of triangle DB is going to be the base DB multiplied by the same perpendicular height en n so if I take a ratio of area of these two triangles the top triangle a d e divided by the area of triangle dbe e what am I going to end up with I&#39;m going to get I basically need to divide these two this divided by this so what&#39;s going to happen will get cancelled if I do a division this guy is going to get cancelled so what will I get a by DB and half will of course get canceled clear so if you&#39;re finding a ratio of the area of these two triangles ad and DB you can see half will get canceled n will get canceled so I&#39;m going to be left with ad / by DB and AD by DB is very important because this is what we looking for in the&#39;s theorem we have to prove a by DB is equal to e AE by EC we have to prove that they are in the same ratio so the main thing what we are doing here we are looking for a ratio of area of triangle a e divided by area of triangle DBC sorry dbe these two triangles so a d the top triang triangle and this triangle if you do the ratio of their areas you will find half will get cancelled the perpendicular height is same because the E is the vertex so you&#39;re dropping the same perpendicular so what you&#39;re left with is AD / DB that&#39;s all you&#39;re left with so that&#39;s great now let&#39;s take a look if we do the same thing but now on this side so instead of considering that is the base if we again look at our top triangle a a d e and this time consider a e as the base so what will my perpendicular height be this time it&#39;s going to be DM so if I consider area of triangle a then with AE as the base the perpendicular height is going to be DM this will be my height H so what will be the area of the triangle ad half time AE uh this should not be EC oh sorry half time a e * DM so you can see area of triangle is going to be half * a e * the perpendicular height DM and what is the area of the triangle below that that is the triangle d e c or ecd whatever you want to call it area of triangle ecd this time this will be the base and the perpendicular height is going to remain the same if you&#39;re looking at triangle a d c because this is the base and this is the perpendicular height same so triangle of uh area of triangle ecd is going to be half times the same perpendicular height DM but this time the base is EC for the lower triangle so again if you divide what&#39;s going to happen half will get canceled if you find the ratio of area of these two triangles half will get canceled DM will get cancelled so we are left with AE / EC and that&#39;s exactly what we want for the theorem AE / EC so now if we compare so here what we have got ratio of area of the two triangles ad by DB here we have got ratio of area of two triangles a by AC we need to prove that they are equal now for them to be equal this area guys need to be equal so what you can see here here we have the same in the numerator we have the same triangle triangle a and in this second equation also we have the same triangle ad but the denominators are different here we have triangle dbe which means we looked at this triangle the one on the left and here we have the triangle ecd the one on the right but if you guys carefully look at these two triangles dbe e and D EC or what did we call it ecd can you guys tell me what is the area of these two triangles are they equal or not so again I&#39;m saying triangle dbe and triangle e CD do they have same area the answer is yes why look carefully if you consider now Ed as the base if you consider the common base you can see that both triangles have the same perpendicular height if you consider this as the base both the triangles have the same perpendicular height because they are between parallel lines so clearly we can see that these two triangles between the parallel lines and formed by these dotted lines are having equal area so therefore what can we say that triangle dbe and ecd are on the same base and between parallel lines so their area is going to be same dbe and ecd have the same area so the denominators are same so that means area of ad by dbe is going to be same as area of ad by ecd since denominators are same therefore we can clearly get a d by DB is equal to AE by EC and so we have proved the tales theorem Tales theorem could also be proved using similar triangles but we did not use that approach we used the area approach it looked a little complicated because we were doing ratio of areas and we were considering a lot of triangles here but in the end you saw that it all worked out same because these two triangles at the bottom have the same area so this one and this guy here both these triangles ended up having same area so a d by DB is equal to AE by e easy we have proved the tales theorem or the basic proportionality theorem so please practice this proof first time you see it it looks complicated but it is all working with the area half base into height the area formula and then considering the correct base and the correct perpendicular that&#39;s it then just writing the ratios you will get the answer now what are some interesting corollaries from the tales theorem so one important coroller is that in if you have a triangle ADB like this where you have de parallel to BC and it intersects you know the other two sides at these distinct points a at d and e then what we can say by theales theorem we saw that a d by DB is equal to AE by EC right we talked about that this a by DB so this parallel line is dividing the two sides in the same ratio is equal to AE by EC but not only that also a d also AB so either you take these two parts a d by DB is equal to AE by e c or you can take the whole line AB / DB is going to be equal to AC the entire length divided by EC okay and similarly you can take AB this entire length AB ided by ad so this time you take only the top part is going to be equal to a divided by AE that means not only you just take a ratio of these two parts you can even take the full length and one part they will all be in the same ratio so all these ratios are going to be equal and you guys can check that with an example so for example if you take like we had taken uh let&#39;s say this one was 2 cm we had taken this as 4 cm this one as 3 cm and 6 cm so let&#39;s confirm so if you confirm the first one a by DB so what is ad by DB we saw that is 2x 4 will be equal to 3x 6 AE by E because we know half is equal to half so this ratio is correct now if you want to prove this one AB by DB what we get here AB is this entire length that means you add it up 2 + 4 so I&#39;m getting 6 divided by DB 4 needs to be equal to AC the entire length is 6 + 3 9 divided by EC which is 6 and now you guys can confirm and check this is going to be 3x2 this is also going to be 3x2 there you go and you can check the last one also so basically you can take either just part of of the side or the entire side but again in the same corresponding way right so here if you take the full length AB it&#39;s going to be 2 + 4 6 divided by just the top part now two and this side is going to be AC the entire length 6 + 3 9 divided by just the top part AE which is three and if you guys check both the 6X 2 is 3 9 by 3 is 3 so you can check with some simple example you can see all these ratios whether you take the part of it all the entire thing they&#39;re all going to be equal so here remember you can remember this with this diagram that if you take this entire length divided by just this part and this entire length divided by just the top part they are going to be in the same ratio so how do you prove this coroller it&#39;s very easy because by theal theorem we know a by DB is equal to a by e now what is the trick is you add one to each side so the trick trick of proving the cor corer is adding one to the left and right hand side when you do that what happens the denominator is going to be DB so the numerator is going to become ad plus DB because the LCM is DB so ad plus DB is what if you guys take a look a plus DB is nothing but the entire side AB so this whole thing if you add it up ad plus DB is nothing but ab and similarly AE plus e c is nothing but AC so you can see so very simple you start off with Tales theorem just add one on both sides and you can easily prove the coroller similarly you can prove the second coroller also you start now not with ad by DB you start with the reciprocals TB by a is equal to EC by AE so you take the reciprocal this time now again you add one and now when you calculate you&#39;re going to get the same thing this entire length AB divided by ad the top part is going to be AC / AE so very simple to prove it you just add one on the left hand side right hand side in one case you start with the tales theorem in another one you start with the reciprocal of the tales theorem done how are we getting AB by a so see we started with DB by a and when you do+ one the LCM is AD so this will go DB + a or you can think 1 is ad by a so the numerator is DB plus a now look at the diagram what is DB + a this entire length AB so I&#39;m replacing this DB + a d with AB similarly I&#39;m replacing AC + AE with AC just simply adding it up now let&#39;s talk about the converse of tales theorem what is the meaning of Converse Converse means the opposite it&#39;s like sort of reversing the theorem so what do we say here if if a line divides any two sides of the triangle in the same ratio that means this time what is given to you let&#39;s say you have a line de and it is dividing these two sides in the same ratio so let&#39;s say this time uh this length is uh let&#39;s say 4 cm and this is 2 cm and here we have something like 6 cm and 3 cm so can you see the line de is divided both these sides in the same ratio because what is a by a / DB is 4x2 which is 2 and similarly AE / EC is nothing but 6x3 which is again two so you can see both are being divided in the same ratio which is 2 is to 1 if this happens then this means that the line that you have drawn de is parallel to BC so if this happens then we know that de is parallel to BC in tal theorem what was it it was opposite it was given that de is parallel to BC and we said that the ratios of these two sides are equal here it is opposite the ratios are given equal the that means based on it the line de must be parallel to the line to this side BC so is the converse or the opposite clear again how do you prove the converse of thales theorem so proof is very interesting let&#39;s take a look so first we&#39;ll write what is the Converse that we have to prove again we&#39;ll start off with the given triangle and what is given to us this is given that ad by DB is equal to AE by e c what do we need to prove that de this guy de is parallel to BC okay we will try to prove this in a very interesting way by the technique known as contradiction contradiction means what we will assume that it&#39;s not true and we&#39;ll prove that&#39;s not possible so it&#39;s like we&#39;re contradicting ourselves it&#39;s like saying you know uh like today is a Wednesday no no today is a Thursday no it&#39;s not a Thursday it&#39;s a Wednesday sort of like contradicting yourself so let&#39;s see what do we do here so what we going to do is we&#39;re going to say let de is not parall to BC so we want to prove D is parallel to BC but we say no no D is not parallel to BC so this is called the contradiction technique so what we want to prove we assuming it&#39;s false so let de is not parallel to BC so if this is not parallel we we going to say no we are going to draw a line which is parallel so we have drawn this random line DF and we are going to say let DF be parallel to BC I know it looks crazy DF doesn&#39;t look parallel to to BC but we&#39;re going to take it right so this is a crazy technique where we say we draw another line which is parallel to BC now if DF is parallel to BC this side BC that means I can apply Tales theorem in it what did Tales theorem say that the ratio of these two sides so a d by DB will equal to AF by FC I can apply th theorem because I drew this magical parallel line so it&#39;s like a Harry Potter line right we did this by Magic we have drawn another line and we are saying by Tales theorem this ratio is true but what is given to us it is given that a by DB is equal to a by EC this was what is given so if you compare these guys the left hand side is the same ad by DB is AF by AC uh AF by FC here ad by DB is a by EC so since the left hand side is same I can therefore say AF by FC is equal to a by EC the right hand sides I&#39;m comparing that means these guys are equal AF by FC is equal to AE by EC clear what we have done till now where we drew a line which was parallel to BC so we&#39;ll start with this AF by FC is equal to a by EC and we&#39;ll do the same trick of adding one on both sides now when you add one something interesting happens the numerator becomes AF by plus FC what is AF Plus FC it is this entire side AC similarly here you get a plus EC so AE plus EC so again you get the entire side AC and now what&#39;s going to happen this AC AC will cancel AC means this side AC not air conditioner this entire side AC and it&#39;s going to get cancelled so what are we left with FC is equal to EC now carefully look at our diagram we are basically saying that this guy FC is equal to EC if FC is equal to AC this can only be possible if E and F are same yes because how can f c equal to EC it is only possible when E and F coincide only then this can be true if they coincide remember we said we had drawn DF parall to BC now if these guys are the same point F and E that means since you drew DF parallel to BC that means de is also parallel to BC and there we have proved the converse of Tes theorem that if the ratios are given equal that means the line is basically de is basically parallel to BC so it is not DF it is basically since they are the same line that means de and BC are parallel to each other so there we have done a lot of theorems the theorem the converse of tales theorem corollaries now let&#39;s get into some exciting sums here&#39;s the first question in the figure de is given parallel to BC so whenever you see this kind of triangle where this uh you know a line in the middle is parallel to this side you know that this is probably going to use Tales theorem or basic proportionality and now what is given to you that a is given as 4x - 3 a is 8x - 7 so all these lengths are given you need to find the value of x so how will you guys do it basically you will apply thales theorem so we are going to say Tales theorem or by basic proportionality theorem by Tales theorem since de is parallel to BC based on this we know that a d D / DB is equal to AE by e c yes it will this parallel line will divide the sides into equal ratios right ad by DB equal to a by e c now just go ahead and substitute so what will we substitute here ad is given as 4x - 3 / 3x - 1 and that&#39;s equal to 8x - 7 / 5x - 3 now how do you solve this equations because you have to solve for x so one simple trick is to cross multiply since there&#39;s nothing much simplification we can do so we can cross multiply here so basically we&#39;ll get 4x - 3 * 5x - 3 = 8x - 7 * 3x - 1 now I guess to solve this you guys will have to expand out the left hand side right hand side and see what equation you get and please solve that equation so can you guys do this for me so please multiply out these numbers and this one and see what you get so if you guys simplify you&#39;ll get something like 20 X2 - 12x - 15x + 9 right so you&#39;ll get all these numbers here and if you further simplify you&#39;re going to get 4x^2 - 2x - 2 which you can divide by 2 so you&#39;re going to get 2x2 - x - 1 = 0 what kind of equation is this quadratic equation because it has x² so you can solve it by middle term breaking Shar achara formula right so always first try middle term breaking and see if you can easily solve it and you look at the factors you know uh 2 and minus1 here so if you break it into - 2x + x you&#39;ll easily get these factors of x - 1 * 2x + 1 = 0 so either X - 1 is 0 or 2x + 1 is 0 which basically means either X is = 1 or x = -2 so let me write this again since it&#39;s not clearly visible so either X is = 1 or 2x + 1 is 0 which means X is - half now why do we say Min - half is not possible can you have two answers for X but Min - half is not possible why- half is not possible because because if you go and substitute min-2 what will you get 4 * -2 is -2 -3 so we getting a length of - 5 that doesn&#39;t make sense and same for the other cases so this is not possible because this will give us negative lengths so the only answer possible therefore X is equal to 1 is the final answer so that&#39;s the only answer which is possible here and you guys can substitute back and check 4 4 - 3 uh so if x is 1 4 - 3 that means this is 1 this is 3 - 1 this is 2 8 - uh 7 this is going to be 1 5 - so 2 so there you can see the ratios are same 1 is 2 1 is 2 that means we have found the correct answer and negative is not possible because it&#39;s going to give you negative lengths of these sides ad DB and so on so always whenever you get a quadratic you get multiple answers please check if your answer makes sense here&#39;s the next question p and Q are points on the sides de and DF of uh DF right this triangle such that DP is 5 cm so let&#39;s write down the values here DP is given to be 5 cm DQ is given as 6 cm de is 15 so this entire length de and qf this length is 18 cm so I know this diagram is not drawn to scale it&#39;s a rough diagram because qf doesn&#39;t look 18 right uh and this D is uh 15 this entire length so the question is is PQ parallel to EF what do you guys think so one is you could use this you know small side is to the big side but uh Tales theorem might be simpler because we want to prove PQ is parallel to EF if you just want to you know calculate this is to this that you can easily find out because what is the length of P 15 - 5 so we can do that the length of p is going to be D minus DP which is nothing but 15 - 5 so that&#39;s 10 cm so we know that our length P here is basically 10 cm so now to prove that if PQ is parallel to EF that means we are using Converse of thees right that if the sides are in equal ratio then it&#39;s going to be parall so let&#39;s find the ratio what is my ratio here DP is to PE so DP by pe is 5 by 10 which is basically half and what is my other ratio here DQ is to qf the other ratio is DQ ided by qf is going to be 6 by8 which is nothing but 1/3 so can you guys see the ratios are not equal so we can see therefore TP is to PE is not equal to DQ is to qf so I cannot apply Converse of thals for Converse of theales these ratios needed to be equal then we could confidently say that DQ is parallel to EF so therefore the answer is going to be therefore uh PQ is not parallel to the side EF and this is the final answer so see that&#39;s how we do it we find the ratios here and you saw the ratios of 1/3 and half so we can clearly say that they are going to be not parallel by Converse of thales theorem let&#39;s look at the next question this is very interesting where we have ABCD is a trapezium with AB parallel to DC and E and F are points on non-p parallel sides a and BC such that EF is parallel to AB so clearly you can see here AB is given parallel to DC in this trapezium and it says that EF is parallel to AB now we know if EF is parallel to AB that means EF will also be parallel to the other uh other side DC what do we need to show that AE is to e is equal to BF is to FC so since you have all these guys parallel can we simply say that this is true by theales theorem is this going to be correct because can I simply say that the ratio of these parts are going to be equal to the ratio of these two parts by Tales theorem am I done yes or no the answer is this is wrong why because reminding you that Tales theorem is for triangles there is no triangle here you are looking at a trapezium so this is not correct Tales theorem cannot be applied to trapeziums so what do we do very simple how about we break this trapezium or this quadrilateral into triangles because Tales theor we can apply for triangles so what you guys can do here go ahead and do a construction let&#39;s say join AC so if you join the diagonals of the trapezium so any one of the diagonal you&#39;re basically breaking it into two parts so there you can see so what you&#39;ll be doing construction where basically you join the DI diagonal AC now you can see you have nice two triangles and here we know that a Ab is parallel to EF and EF is also parallel to DC so you can write those points they&#39;re all parallel now if you look things become easy because the ratio of AE is to Ed so let&#39;s say this meets diagonal meets this guy at what should we call it let&#39;s say this is uh EF we can call this G so that means AE is to Ed D is going to be equal to AG is to GC by the&#39;s theorem because you can see these two parallel lines so now we can say by theales theorem if you&#39;re looking at triangle a DC so in Triangle ADC by theales theorem what do we know AE is to Ed AE by Ed is going to be equal to AG by GC so this we know the ratios of this is to this is going to be equal to this is to this now you focus on the other triangle triangle ABC in triangle ABC again this is parallel to this so that you should definitely write that all these three sides are parallel so now looking at triangle ABC again by thales theorem so what do we have we can say AG is to GC is equal to BF is to FC there you go so again we can say AG so this side AG is to GC is equal to BF is to FC because you don&#39;t always have to start from the top right you can take this is to this equals this is to this so there you go now you just equate these two so we&#39;re going to equate 1 and two so from 1 and two we can clearly see AG is to GC is common part so a by Ed will equal to PF by FC so we can say from 1 and 2 a e by e d is going to be equal to BF by FC there you go we are done proved so simple so the whole trick was breaking it into two triangles because you can&#39;t apply thales theorem to a quadrilateral like a trapezium so there you go okay here we have labeled this point as M or whatever you want to call it and there you can see the same same proof using the theorem you could also have used the diagonal DB you&#39;ll get the same thing so there you can see using both the things uh theil theorem on both the triangles so this is your triangle one and your triangle two you will get the same answer let&#39;s try another interesting question in the figure de is parallel to BC ad is given 3 cm so again in Geometry make sure you&#39;re marking everything here so a is 3 cm AE is 2 cm so this length is 2 cm BD is 2 cm so BD is 2 cm de is 6 cm and what do we need to find the length of BC this is the unknown so how do you guys plan to do this and you have been given that de is parallel to BC that we have marked in the picture here so can we find this side by applying Tales theorem can we say you know this by this equals this by this or how do we do it can we apply theales theorem for this one so Thal theorem according to thales theorem we know that if this de is parallel to BC so by theales theorem we know that when you have de parallel to BC then a is to DB the ratio of a by DB will equal to AE by EC because this parallel line divides these Parts into equal ratio but the problem is we don&#39;t want to find the length AC we want to find the length BC so can we apply Tales theorem here the answer is no so here basically Tales theorem or basic proportionality theorem does not help so what will you do here so because to find thales theorem doesn&#39;t help you find the length of the parallel side it talks about these the nonp parallel side right right the parallel line dividing the other two sides so how do you solve this so one simple trick you can do here when you can&#39;t apply Tales theorem again look back at the triangles because now you&#39;re looking at What triangle a and you might want to look at triangle ABC because you want to find the length BC what do we know about these triangles since the lines are parallel we know that this angle is equal to this angle by corresponding ing angles again you have parallel lines these angles are equal and of course this angle is common angle a so we know that these two triangles the top triangle ad and the bottom triangle ABC are similar so when you can&#39;t apply THS theorem go to the similar triangles so we can say that triangle a d e is similar to triangle ABC by a a or AA a angle angle criteria so now once they are similar you know you can find the ratio of the corresponding sides are equal that will help you find the length of BC so let&#39;s go ahead and do that so since they are similar to each other I can clearly say that a so the length ad is to the whole uh the other bigger triangle ad by AB is equal to D / BC corresponding parts of similar triangles are in the same ratio right these corresponding sides they are in the same ratio so this is what we do now you just go ahead and substitute the values here very simple ad is 3 cm AB is given to you the whole length AB will be 3 + 2 so carefully add it up this is going to be 3 + 2 which is 5 the length d uh the length D is given to you 6 cm so we&#39;ll subtitute that and we need to find the length BC so please go ahead and solve this so basically we are getting BC is 6 * 5 / 3 or BC = 6 * 5 / 3 which is nothing but 10 cm that&#39;s the final answer here so there you can see that using this similar triangle so what did we do so since Tales theorem did not work using similarity so similar triangles came to the rescue here since we wanted to find this side BC and you can see you&#39;ll get the answer very easily and so see by the a criteria you must show that the triangles are similar and we got the same answer 10 cm so hope the tales theorem or the basic proportionality theorem and its Converse is crystal clear to you now and we practice some very very interesting and important questions and do check out the other courses on our website we have physics chemistry biology and Maths for cbsc Class 8 9 and 10 so guys if you haven&#39;t taken the other subjects do take them and do share it out with your friends for ICC students once again we have physics chemistry biology and Maths for classes 8 9 and 10 and if you want to learn coding we have Python Programming we have Java coding both are great languages to learn computer programming and we have physics and chemistry for the Cambridge IGCSE which is the international board so please do share out our courses with your friends make sure you have subscribed to our YouTube channel we also have a Hindi channel it&#39;s called Manoa Academy Hindi to stay connected with Manoa Academy and keep learning

Transcript for:Understanding Thales Theorem and Its Applications

Transcript for:
Understanding Thales Theorem and Its Applications