Geometry Class: Similarity of Triangles

[Music] hi everyone this is Ravi Prakash and welcome to seventh class or geometry right so here we'll discuss about similarity of triangles right similarity of triangles so in similarity of triangles right how to prove two triangles are similar what are its use how to apply it right everything will look right similarity of triangles see how to prove two triangles are similar right three ways basically first is SSS second is a a similarity and third is SAS similarity right there are three ways by which we can prove two triangles to be similar right notice SSS similarity so SSS similarity basically in two triangles in two triangles ratio of ratio of all the sides all the pairs of sides are same okay into triangles ratio of all the corresponding bonding pairs of sides are same pairs of sides are same or equal for example if here I have a triangle called let's say 6 11 and 13 the triangle ABC and then I have a triangle here like PQ are in this range the sides are three five point five and six point five so I can say that these two triangles are similar I can write triangle ABC is similar to triangle PQR right by SSS similarity why because see the ratio is same now 6 upon 3 is 2 ok 11 upon 5 point 5 is also 2 and thirty nine point six point five is also 2 right so all these corresponding pairs of ratio are same right a B upon its length it's like a b upon a b upon the P cube is equal to BC upon Q R is equal to AC upon PR a C upon PR solar same right there so it is similar okay so second one is a a similarity fine now what is a similarity a a similarity is when two angles are equal when two angles are equal right so two angles of obviously when comparing two triangles right so when comparing in two triangles both the angles are equal that is a a similarity okay now let's take example and bring it a red triangle ABC okay triangle ABC ABC and let's end drawing a line here that is de and de is parallel to BC okay de is parallel to BC right so I can write here that there are two triangles triangle gay de and triangle ABC now I can write triangle a de is similar to triangle ABC why it is similar because of a a similarity that is sign of similarity why it is similar because of a a similarity see here this angle and this angle will be equal this angle and this angle will be equal right because their corresponding end is fine what a corresponding angle C remember we discussed chivalry basic video this is in correspondent there are two parallel lines when there are two parallel lines the transversal this angle and this angle are equal it's called corresponding angle red and this angle and this angle are equal they are called alternate and is they're called alternate angles right kita and theta these are alternate angles here alternate and this here corresponding angle is fine so two angles are equal so I can say it is similar by a a similarity by a similarity right okay now let cuz Casey what happens when these two triangles are similar right so when two triangles are similar that basically means that okay I can write this treatment here when two triangles are similar it means that the ratio of their ratio of corresponding sides their ratio of corresponding sides is equal to corresponding side sides are equal the ratio of corresponding sides are equal and and ratio of corresponding sides is equal to ratio of heights ratio of heights is equal to ratio of medians is equal to ratio of in radius is equal to ratio of circumradius right very important point when two triangles are similar their ratio of corresponding sides are equal and the ratio of I should write du I should write to 0.2 0.7 when two triangles are similar first point okay ratio of corresponding sides are equal and ratio of corresponding sides is equal to ratio of heights of two triangle is equal to ratio of medians of two triangles is equal to ratio of in radius is equal to H ratio of circle it is red all these will be equal to when two triangles are similar fine okay now see second point in this in this the second point will come let me write here second point so when two triangles are similar it is a first point and second point is reach the ratio of area of two triangles ratio of area of two triangles is is a square of the square of the ratio of corresponding sides and then Heights right because corresponding sides equal to ratio of Heights and all so all this will be equal right now I can easily see why the second point occurs right the strain wise is why the second point occurs it is because area is equal to half into base into height right so now once you take the area of two triangles once you take the area of two triangles right so those area of two similar triangles so half into base into height right it's a b1 h1 it'll let it be b2 h to half of gets cancel so b1 h1 upon b2 h to right so each term will be what now this is base right what is base here base is any set aside of side of the triangle so once it is a side ratios aside into side rate be 1 upon B 2 into H 1 by H 2 this side it becomes what will become square write something side ratio or equal right side when two triangles similar this is equal rate ratio of base and ratio height see ya ratio of corresponding sides will be equal to ratio of heights right so when two triangles are similar the base ratio and height ratio B what is base one of the sides only right one of the sides so base reissue and hydration equal right it will become what it'll become square you become is square right that's why it will become a square base this third equal so multiplication is a right it becomes a square right that's why if the ratio is what in area ratio of area of 210 in his word a square of the ratio of the corresponding sides why square because base ratio and height ratio both are equal in two similar triangles certainly become square in multiplication right so very tenth point about similarity okay with no questions right first we'll do the third one then we'll do questions right third one third one is by SAS similarity as a s similarity now what is sa similarity so SN similarities basically ratio and ratio of two sides when ratio of two sides and the included angle are equal are equal right it has to be it has to be an included angle obviously right see you know making this road we can bring it to angle she'll make it to angle ABC and wicked angle PQR so go this is 10 this is 8 this is 5 this is 4 right I can see this ratio is equal right I can see this a B upon P Q is equal to BC upon Q R is equal to BC upon Q R right so when a B by P Q equal to BC by Q R so in that case this is what this is ratio of sides s and s Elton s and s is done right ratio of two sides first and second are equal now included angle has to be coded that means if this is 30 degree this also has to be 30 D read only included angle whichever two sides you are taking ten and eight and five and four so angle between them so this is a meaning of introvert angle it does to be equal it is equal the regulated okay triangle ABC is similar to triangle PQR by SAS similarity okay so this is the third similarity right this is a third similarity here okay fine now C will discuss few good questions of similarity right it's a very important topic similarity okay and we're doing lots of questions similarity see how to write the ratio of sides and all in similarity right so this will make you clear this will make you clear see if I am taking here that diagram again I'm making it is triangle ABC and this BC is parallel to de so by now we know that triangle ABC is similar to triangle a de vie similar because it is by Philips it vice is similar but because it it is by a similarity right it is by a a similarity so a is in similarity means this angle and this angle are equal corresponding angle this angle it is angular equal right corresponding angle is Theta this theta this Phi this is Phi okay and this angle a is the common angle right angle a is the common so how to write a ratio of sides now right see diagram sometime diagram will be easy sometime diagram will be difficult right here you can directly say okay de upon BC or ad upon a B but in tougher diagrams like in next one you can't simply write it right so in this case I just suggest you to follow the angle voice right you pick any angle suppose you pick the common angle first so in common here in triangle l de common angle opposite is de and in triangle ABC common angle opposite is BC so my first ratio ratio is what de upon B see this is my first pressure right just by common engine so here common angle opposite de here common angle it would be C right now you choose you choose this right that which triangle to take in numerator and which brand is to take in denominator right suppose in this case I have taken triangle a de in numerator so I have taken triangle a de he numerator so I'll take it throughout and triangle ABC in denominator so I'll take it throughout right a lot I don't reverse in between right so common angle opposite de common angle would be C is it in Crandall a de and a B C respectively right now again in numerator and writing triangle hey de now computer angle IDI in triangle a de you pick theta right suppose then strangled big is Theta in goes it wrong again so in triangle a de since we are writing it in numerator a triangle a de theta opposite is a D okay any triangle ABC theta poet is what EB so kdd upon AV right now now it the angle phi phi will pick right so again hinter angle a de since we're at a numerator so angle a de file opposite is AE and triangle ABC Otis fire posit AC so I'll write a e upon AC fine not this not only yes right you remember ratio of corresponding sides is equal to ratio of heights also right so if I can draw us a draw height here you draw a height here so let the height of this triangle is AF and height of this triangle is AG right so this will also be equal to this will also be equal to AF upon a G a F upon a G right so this is the this is how you would write for similarity in trans similarity of triangles right now if I have to take the ratio of also in account side so what I wanted was okay the second point I'm talking about for a second point in a second point was if I take okay what is the ratio of ratio of area of triangle a de upon area of triangle ABC what is the ratio a de upon ABC right now it is ad here so in theory of triangle a de so when two triangle similar second point right laughs to our slide when two triangles are similar the ratio of area is equal to what square of the ratio of corresponding sides and height and all right because I told you what is area AJ is nothing but base into height so a1 is equal to half B 1 H 1 a 2 is equal to half b 2 h 2 4 second triangle right b 2 h 2 for second Prandtl we will take the ratio here take the ratio here this half of will get cancelled right now B 1 by B 2 now this what this is B 1 by B 2 so a 1 by a 2 is equal to B 1 by B 2 into H 1 by H 2 right so since this both are equal so it is the area area ratio sides is going right because both are equal now because when sites triangles are similar both the ratio are see both ratios are equal AE by AC is equal to a by huh so what is this a ratio of what is ratio base here ratio basis what de upon bcc c d e upon bc is equal to AF upon AG so these two are equal okay these two are equal so what will happen when these two are equal so both will be multiplied so it'll become a square right so that's why area of triangle a de to area of triangle ABC for the ratio is this square now de upon BC square is equal to here ad upon a B square is equal to AE upon AC square and then moreover the ratio of heights and holes will go on right so a ratio of area of two triangles is a square of the ratio of the corresponding sides and hydron all right so very important point this one okay now move to next slide see lots of questions are based in this right let's discuss a point here what if in case of right angle triangle this is writing you can read so see we will discuss separately actually right angle triangle similarity but just to make you understand how to solve how to write in case of right angle triangle how to write the ratio because it is quite confusing right so let us say triangle ABC this is a right angle triangle ABC now BD is a perpendicular BD is a perpendicular BD is a perpendicular to trash to side AC so now here we have three pairs of sides right here we have three pairs of sides now let's say here can I write triangle a B so here are three pairs of triangles here not side sorry triangle abd so here your triangle abd can I say triangle abd is similar to triangle BDC can I say it to smaller triangle right triangle abd it's similar to triangle BDC now I can't say it now because I need at least a way to prove it right but here only aim is for two angles have to be correct but in comparing these two triangles only one angle seems to be cool that is this is 19 this triangles is 19 the strength alone uh thing is okay so I need one more and right so that you can easily do just one angle theta you need to consume for you to assume any one angle as theta right what will happen now this is 390 this is Theta this becomes 90 minus theta why because some have to be 180 degree so 1 this is theta is 90 what is this angle 180 minus 90 plus theta so it becomes 90 minus theta right so one angle is ninety one angle is Theta third will be 90 minus theta to make this sum to be 180 degree okay now here again it is 90 this whole is 90 right this whole is 90 and this is 90 minus theta so again this has to be theta okay this is Theta this is Theta this is 90 so this becomes what again 90 minus theta right so just you assume one angle is Theta and you can now solve everything right so you can see that now these two triangles are similar by a similarity because all these three and you'll see a synonym is what if two angles are equal obviously third one has to be equal because it is it is what it is sum has to be 180 degree okay so here also all three angles are equal that means with that means what attitudes triangles are similar so abd is similar to triangle BDC correct so if I write the ratio you write write the ratio you how do I D ratio now you can write D ratio you quickly again by picking angle right you will decide which triangle to keep in numerator so let's say angle a triangle a we need to keep in numerator and triangle BDC to keep in denominator right okay so now we can solve now in numerator triangle a way to pick any end and suppose you pick theta at first so here theta opposite is ad here theta opposite is ad and here theta poet is what BD so I can write the ratio as AD upon BD okay next one you get picnic angle is 90 suppose right so here in triangle abd 90 opposite is AV and in triangle BDC 90 upon it is BC so KB upon VC try it this is how you write the ratio in every question based on the angles right when you prove by a similarity you must have mark the angles right so use that mom tengas for adding the ratio right no other method simply didn't directly do it right third one let's say let's say third one I bigger is 90 minus theta so here 90 minus theta opposite is BD here 90 minus theta pulled is what DC so my ratio is what B D upon DC fine so I can combine these two and from a result result we'll discuss later also so you simply write here BD square is equal to 80 into DC right so like this you can have any kind of any or any pair of triangle you can take right this is the first pair I've taken now second pair is also similar right that means have taken fin first one I have taken two smaller triangles this one's one right now second one I can also take the bigger triangle triangle ABC is also similar to triangle BDC again by this is equation 2 I can write a similarity and again I can write D ratio right same way because for both these triangles be bigger and is monitoring this all the angles are what equal right now again sudden third also generate the bigger one triangle ABC is similar to a smaller one again triangle ADB angle ADB right so this is a very important case of right angle triangle inside a right angle triangle right will again discuss it later right we'll come back to it okay so to find all the values right so this three are very important of similarity because when you write a right angle triangle in a right angle triangle all the three pairs of trainers are similar just two smaller one one bigger one and one lower is smaller one one bigger one and one upper smaller one right all three pairs are similar right and so by similarity can you remember is it also yes right and remember is that here you will come back to it but it's okay more you revise it more you will learn quickly right to find sides in triangle right now let's say this is a this is B and this whole is C this hole is see so it's a result of similarity I am writing here it's a result of similarity this hole is see right the result is actually if if this is a this is B this side is this side this is smaller one I could write a b c and d this ad is nothing but a square by C this C D is nothing but B Square by C and this B D is nothing but a B by C okay is reverse result as a result of similarity only again all these three sites three purifying triangles will put similar handle get this result very point is out right indirectly right this is a this is B this is C this is very easy to remember also this is a square by C this is right ad and red double ad ad s ad is a square by c BD is sorry C D is v square by C so you write here a you right here bu right here C upper one is ad is a square by C lower one BD BD is B square by C and so it not be d CD CDs B square by C and his BD is what this PD is a B by C right so very quickly can get this value for example we have questions also like this right suppose is a triangle here and triangle is sides are 6 8 and 10 ABC 6 8 and 10 and this BD is the in height of the triangle right so what is the value of ad what is the value of D C what is the value of BD right so in one go you can read in one go you can write write what is ad what is the ad here ad is basically a square by C what is a square six square so here it is a t-bird 6 square by 10 that is 3.6 this is ad so ad is 3.6 right what is C de Sade's B square by C what is B Square basis is B right so 8 is square by 10 that is 6 point 4 it says this is BD so this is CD CD but is BD a bit is a B by C but if I give you I see 6 into 8 by 10 that is 4.8 therefore I can write BD is what we D is 4.8 right so not at all time taking simply you write this value get this answer anytime okay it's a very good concept remember it okay by similarity and again we'll come back to similar active adding a triangle again okay so for now what I can write here is okay this to a question you know just do a question here and the question is there is a there is a a square inside a triangle okay liquid rotor angle here there is a square inside a triangle like this okay this a square inside a triangle okay and sides of a triangle are basically twenty thirty four and forty two this is 42 this is 20 and this is thirty-four right this whole side a b c fine find a the question is find a length of side of a square side of a square right again a super cushion which can be solved by similarity in a very easy way right if I understand this right it is a super application of similarity see now when you have figure like this here so I know they did not get all the hidden facts sort of the figure right it all the hidden Friedan facts out of the figure now since it is a square so square means what opposite sides are parallel so this is parallel to this obviously now name it de we can name it de here so now de is parallel to BC so again same thing this will be corresponding angle and this will also be corresponding angle that means I can write that triangle ABC is similar to triangle we see okay so it is similar so I can again write in our issue okay write the ratio how to read e ratio here so now angle a is the comma and the third angle right so you see if you pick mark this as theta you mark this theta theta so modulus Phi and Phi right you see here I home NT side of a square right 12 side of the square ladies side of a square ba later this side of a square by a so all four sides will be equal to the square means all four sides equal right so I want a so I should pick which and then I should not pick theta here because theta opposite here is Haidee and it is not known and neither I am interested also write Phi opposite here is a again AE is not known right what I'm interested in is I need to find the side of a square right so I should pick what I should pick angle a here here so that is a common angle in both a triangle right so here the common angle opposite is a in smaller triangle in bigger triangle common angle a poet is what 42 so I should write a upon 42 okay should write a upon 42 okay no you should also define your red cape on for it equal to I I need to find a value of a right now equal to what equal to what right so this is equal to what right so this is again I need four other angle right so again I told you if I pick the common angle it is fine but again if I take if I pick fire posit fire point a is not known already a is one unknown variable another one unknown AE will come or if I pick theta another unknown ad will come so that means I can't pick so ratio from ratio of corresponding sides right so remember the statement when two triangles are similar ratio of corresponding sides is equal to ratio of height and it told you earlier when all three sides of a triangle are given I can get the reach I can get the value of height right now by equating area by equating area right so same thing I should do here also okay let me rub this part so basically I should write to the ratio of heights I should write to the ratio of heights right now to rate ratio of height here so ratio height can be written let me draw a height here draw I tried this height for both the triangles fine a F and a G okay how did you how did at the height of triangles see height of triangle you can get by two method first one is by area okay first one is by area okay see so what is the area of triangle root under s into s minus a into S minus B into s minus C s minus B into s minus C when necessary perimeter right what is area here so area is what is s here s is 20 plus 34 54 plus 42 96 s s 96 by to pretend this is what I do 48 so 96 by 2 is 48 so s is 48 here correct s is 48 here ok now 48 into S minus a 48 into s minus a right what is s why I say 48 minus 4 T 2 is 6 now s minus B 48 minus 34 is 14 s minus C 48 minus 20 is 128 now what is the area so area a little bit by now see you have got two multiples of 7 year right that means we have got see what I can do here I can split this 6 as 2 into 3 so this 2 with 14 will become 28 so out of 228 128 will come out right out of 228 128 will come out ok what is left here now so 3 is left so 3 into 48 is what 144 3 into 48 is 144 and what is square root of 144 12 so area is what area is 28 into 12 that is 280 plus 56 that is 333 36 this is the area of trundle ok and this I should equate with area of triangle is equal to 1/2 into height into base right what is area 3 3 6 what is height here you need to calculate what is base here for height and base here 42 right so you get the value of a solidity value here simply cancel eight times right so height you will get as 16 height you will get a density right so by this we can get hide as 16 okay now this is the first treasure to get the height right obviously should do the concept right no there's a shortcut also and the super shortcut right if you start using Pythagorean triplets now you will find gym it really easy right because 30 to 45 no question you enjoy solving because of Python Spicer interpret right so in second method I'll use PI 0 temperate right remember that I heard I got height as 16 okay I got high death 16 fine height as 15 I got okay now by use of second method by use of Pythagorean triplet okay so how to use the power to use Python interpreter see method to five Pythagorean triplet see the triangle here triangle is 20 34 and 42 I am calculating height of the triangle try it now you just get a triplet we just did a triplet triplet with hypotenuse 34 right so just cross-check all the triplet right because 34 can be in any triplet also or it can be derived right can it can be multiple of an interpolated so third the as soon as you say 34 it strikes that it is a double of 17 and double of 17 means what the relative order plate is triplet is what triplet is eight to fifteen seventeen okay how again 34 I'd look at 34-year again you cross check those six basic I told her to remove those six basic triplet right 3 4 5 5 12 13 7 24 25 and 850 79 40 41 right so all those you cross check here and you see that 34 34 is a hypotenuse it is a multiple about 17 then it all the hypotenuse you write 3 4 5 5 is the hypotenuse 34 is not a multiple of 5 5 12 13 again 30 is the hypotenuse 34 not equal to 13 now similarly 24 25 25 the hypotenuse 34 is not a multiple of 25 then 8 15 17 right 34 is a multiple of 72 times right so if a triplet is 8 15 17 and it's double of it it develop it right so should I should write okay double is what 8 15 17 is 34 what a double of it never is 16 30 and 34 double is 16 30 and 34 right now I can't write I can't write this side as 30 why because in a triangle this is a triangle this is a triangular in a triangle hypotenuse is the longest side okay and 20 is the hypotenuse it is a longest side right so in this side can't be 30 so 16 and 30 so she should be 113 should be this one and it should be 16 okay so this is a fundamental iterate again make it a little bit of complications in the diagram where I did let me make it again okay just concentrate a very good concept here to save you time 20 34 42 this is the height so it should be what it should be 8 15 17 devil I told you that is 16 30 and 34 now not always too right so we know that this is this side also it should satisfy because this is also right angle triangle so it should satisfy this because I have made a bit of guesswork right so just check cross shape is satisfying or not it is satisfying C is 42 this 30 so it should be - all right this is writing with triangles they said also adding with language at 12 16 20 and under triplet right derived from 3 4 5 3 4 5 into 5 times okay so I 3 4 5 into 4 times 2x sufficient when this is at satisfying right this satisfying what is the height so height is 16 so height we got a 16 right stretch try see you can always do it this but it is a concept right but try this kind of logic right you should try this kind of logic and you can read the quotient within minutes right this these things are very important very very important right so try using these things okay triplets triplets engage a right angle triangle is a super concept right you should always pray by putting triplet try to recognize it or play it right you can save a lot of time here okay so anyway I got height as 16 right I got height as 16 okay now come back to this question having all this right now height is 16 okay so that means what basically now the ratio sites of toot - triangle right so common angle ability is a common angle opposite is 21 right common angle a poet is a common angle suppose it is nothing but 21 okay this is a height of triangular so 21 right now no so ratio of two's ratio of sides of triangle is equal ratio of height ratio of idea so then you see this is a this is a right this is a that means it's a straight line is a square is ninety degree three also 90 degree so straight line that this also should be a this also should be a so this is a hole is 16 so high it off a smaller triangle how much 16 - a pride 16 this length is a this length is a this FG is a right FG is a why because parallel to the side of a square okay so a holy 16 so height of a smaller triangle is 15 minus a upon height of bigger triangle that is 16 right so this is the ratio here we got here it's a super question letting get the value of a from here 16 a is equal to 42 into 16 right 420 + 252 that is 672 672 - 42 42 K right that means 58 a is equal to 672 therefore a is equal to 672 by 58 okay that is 336 by 29 so answer is what 336 by 29 this is the answer for this question what is the value of a 336 by 29 okay so very nice question lots of concerts record is this video right especially this question I understand this question you can apply in many concept many other question right this concept okay thank you we'll continue next video [Music]

[Music] hi everyone this is Ravi Prakash and welcome to seventh class or geometry right so here we&#39;ll discuss about similarity of triangles right similarity of triangles so in similarity of triangles right how to prove two triangles are similar what are its use how to apply it right everything will look right similarity of triangles see how to prove two triangles are similar right three ways basically first is SSS second is a a similarity and third is SAS similarity right there are three ways by which we can prove two triangles to be similar right notice SSS similarity so SSS similarity basically in two triangles in two triangles ratio of ratio of all the sides all the pairs of sides are same okay into triangles ratio of all the corresponding bonding pairs of sides are same pairs of sides are same or equal for example if here I have a triangle called let&#39;s say 6 11 and 13 the triangle ABC and then I have a triangle here like PQ are in this range the sides are three five point five and six point five so I can say that these two triangles are similar I can write triangle ABC is similar to triangle PQR right by SSS similarity why because see the ratio is same now 6 upon 3 is 2 ok 11 upon 5 point 5 is also 2 and thirty nine point six point five is also 2 right so all these corresponding pairs of ratio are same right a B upon its length it&#39;s like a b upon a b upon the P cube is equal to BC upon Q R is equal to AC upon PR a C upon PR solar same right there so it is similar okay so second one is a a similarity fine now what is a similarity a a similarity is when two angles are equal when two angles are equal right so two angles of obviously when comparing two triangles right so when comparing in two triangles both the angles are equal that is a a similarity okay now let&#39;s take example and bring it a red triangle ABC okay triangle ABC ABC and let&#39;s end drawing a line here that is de and de is parallel to BC okay de is parallel to BC right so I can write here that there are two triangles triangle gay de and triangle ABC now I can write triangle a de is similar to triangle ABC why it is similar because of a a similarity that is sign of similarity why it is similar because of a a similarity see here this angle and this angle will be equal this angle and this angle will be equal right because their corresponding end is fine what a corresponding angle C remember we discussed chivalry basic video this is in correspondent there are two parallel lines when there are two parallel lines the transversal this angle and this angle are equal it&#39;s called corresponding angle red and this angle and this angle are equal they are called alternate and is they&#39;re called alternate angles right kita and theta these are alternate angles here alternate and this here corresponding angle is fine so two angles are equal so I can say it is similar by a a similarity by a similarity right okay now let cuz Casey what happens when these two triangles are similar right so when two triangles are similar that basically means that okay I can write this treatment here when two triangles are similar it means that the ratio of their ratio of corresponding sides their ratio of corresponding sides is equal to corresponding side sides are equal the ratio of corresponding sides are equal and and ratio of corresponding sides is equal to ratio of heights ratio of heights is equal to ratio of medians is equal to ratio of in radius is equal to ratio of circumradius right very important point when two triangles are similar their ratio of corresponding sides are equal and the ratio of I should write du I should write to 0.2 0.7 when two triangles are similar first point okay ratio of corresponding sides are equal and ratio of corresponding sides is equal to ratio of heights of two triangle is equal to ratio of medians of two triangles is equal to ratio of in radius is equal to H ratio of circle it is red all these will be equal to when two triangles are similar fine okay now see second point in this in this the second point will come let me write here second point so when two triangles are similar it is a first point and second point is reach the ratio of area of two triangles ratio of area of two triangles is is a square of the square of the ratio of corresponding sides and then Heights right because corresponding sides equal to ratio of Heights and all so all this will be equal right now I can easily see why the second point occurs right the strain wise is why the second point occurs it is because area is equal to half into base into height right so now once you take the area of two triangles once you take the area of two triangles right so those area of two similar triangles so half into base into height right it&#39;s a b1 h1 it&#39;ll let it be b2 h to half of gets cancel so b1 h1 upon b2 h to right so each term will be what now this is base right what is base here base is any set aside of side of the triangle so once it is a side ratios aside into side rate be 1 upon B 2 into H 1 by H 2 this side it becomes what will become square write something side ratio or equal right side when two triangles similar this is equal rate ratio of base and ratio height see ya ratio of corresponding sides will be equal to ratio of heights right so when two triangles are similar the base ratio and height ratio B what is base one of the sides only right one of the sides so base reissue and hydration equal right it will become what it&#39;ll become square you become is square right that&#39;s why it will become a square base this third equal so multiplication is a right it becomes a square right that&#39;s why if the ratio is what in area ratio of area of 210 in his word a square of the ratio of the corresponding sides why square because base ratio and height ratio both are equal in two similar triangles certainly become square in multiplication right so very tenth point about similarity okay with no questions right first we&#39;ll do the third one then we&#39;ll do questions right third one third one is by SAS similarity as a s similarity now what is sa similarity so SN similarities basically ratio and ratio of two sides when ratio of two sides and the included angle are equal are equal right it has to be it has to be an included angle obviously right see you know making this road we can bring it to angle she&#39;ll make it to angle ABC and wicked angle PQR so go this is 10 this is 8 this is 5 this is 4 right I can see this ratio is equal right I can see this a B upon P Q is equal to BC upon Q R is equal to BC upon Q R right so when a B by P Q equal to BC by Q R so in that case this is what this is ratio of sides s and s Elton s and s is done right ratio of two sides first and second are equal now included angle has to be coded that means if this is 30 degree this also has to be 30 D read only included angle whichever two sides you are taking ten and eight and five and four so angle between them so this is a meaning of introvert angle it does to be equal it is equal the regulated okay triangle ABC is similar to triangle PQR by SAS similarity okay so this is the third similarity right this is a third similarity here okay fine now C will discuss few good questions of similarity right it&#39;s a very important topic similarity okay and we&#39;re doing lots of questions similarity see how to write the ratio of sides and all in similarity right so this will make you clear this will make you clear see if I am taking here that diagram again I&#39;m making it is triangle ABC and this BC is parallel to de so by now we know that triangle ABC is similar to triangle a de vie similar because it is by Philips it vice is similar but because it it is by a similarity right it is by a a similarity so a is in similarity means this angle and this angle are equal corresponding angle this angle it is angular equal right corresponding angle is Theta this theta this Phi this is Phi okay and this angle a is the common angle right angle a is the common so how to write a ratio of sides now right see diagram sometime diagram will be easy sometime diagram will be difficult right here you can directly say okay de upon BC or ad upon a B but in tougher diagrams like in next one you can&#39;t simply write it right so in this case I just suggest you to follow the angle voice right you pick any angle suppose you pick the common angle first so in common here in triangle l de common angle opposite is de and in triangle ABC common angle opposite is BC so my first ratio ratio is what de upon B see this is my first pressure right just by common engine so here common angle opposite de here common angle it would be C right now you choose you choose this right that which triangle to take in numerator and which brand is to take in denominator right suppose in this case I have taken triangle a de in numerator so I have taken triangle a de he numerator so I&#39;ll take it throughout and triangle ABC in denominator so I&#39;ll take it throughout right a lot I don&#39;t reverse in between right so common angle opposite de common angle would be C is it in Crandall a de and a B C respectively right now again in numerator and writing triangle hey de now computer angle IDI in triangle a de you pick theta right suppose then strangled big is Theta in goes it wrong again so in triangle a de since we are writing it in numerator a triangle a de theta opposite is a D okay any triangle ABC theta poet is what EB so kdd upon AV right now now it the angle phi phi will pick right so again hinter angle a de since we&#39;re at a numerator so angle a de file opposite is AE and triangle ABC Otis fire posit AC so I&#39;ll write a e upon AC fine not this not only yes right you remember ratio of corresponding sides is equal to ratio of heights also right so if I can draw us a draw height here you draw a height here so let the height of this triangle is AF and height of this triangle is AG right so this will also be equal to this will also be equal to AF upon a G a F upon a G right so this is the this is how you would write for similarity in trans similarity of triangles right now if I have to take the ratio of also in account side so what I wanted was okay the second point I&#39;m talking about for a second point in a second point was if I take okay what is the ratio of ratio of area of triangle a de upon area of triangle ABC what is the ratio a de upon ABC right now it is ad here so in theory of triangle a de so when two triangle similar second point right laughs to our slide when two triangles are similar the ratio of area is equal to what square of the ratio of corresponding sides and height and all right because I told you what is area AJ is nothing but base into height so a1 is equal to half B 1 H 1 a 2 is equal to half b 2 h 2 4 second triangle right b 2 h 2 for second Prandtl we will take the ratio here take the ratio here this half of will get cancelled right now B 1 by B 2 now this what this is B 1 by B 2 so a 1 by a 2 is equal to B 1 by B 2 into H 1 by H 2 right so since this both are equal so it is the area area ratio sides is going right because both are equal now because when sites triangles are similar both the ratio are see both ratios are equal AE by AC is equal to a by huh so what is this a ratio of what is ratio base here ratio basis what de upon bcc c d e upon bc is equal to AF upon AG so these two are equal okay these two are equal so what will happen when these two are equal so both will be multiplied so it&#39;ll become a square right so that&#39;s why area of triangle a de to area of triangle ABC for the ratio is this square now de upon BC square is equal to here ad upon a B square is equal to AE upon AC square and then moreover the ratio of heights and holes will go on right so a ratio of area of two triangles is a square of the ratio of the corresponding sides and hydron all right so very important point this one okay now move to next slide see lots of questions are based in this right let&#39;s discuss a point here what if in case of right angle triangle this is writing you can read so see we will discuss separately actually right angle triangle similarity but just to make you understand how to solve how to write in case of right angle triangle how to write the ratio because it is quite confusing right so let us say triangle ABC this is a right angle triangle ABC now BD is a perpendicular BD is a perpendicular BD is a perpendicular to trash to side AC so now here we have three pairs of sides right here we have three pairs of sides now let&#39;s say here can I write triangle a B so here are three pairs of triangles here not side sorry triangle abd so here your triangle abd can I say triangle abd is similar to triangle BDC can I say it to smaller triangle right triangle abd it&#39;s similar to triangle BDC now I can&#39;t say it now because I need at least a way to prove it right but here only aim is for two angles have to be correct but in comparing these two triangles only one angle seems to be cool that is this is 19 this triangles is 19 the strength alone uh thing is okay so I need one more and right so that you can easily do just one angle theta you need to consume for you to assume any one angle as theta right what will happen now this is 390 this is Theta this becomes 90 minus theta why because some have to be 180 degree so 1 this is theta is 90 what is this angle 180 minus 90 plus theta so it becomes 90 minus theta right so one angle is ninety one angle is Theta third will be 90 minus theta to make this sum to be 180 degree okay now here again it is 90 this whole is 90 right this whole is 90 and this is 90 minus theta so again this has to be theta okay this is Theta this is Theta this is 90 so this becomes what again 90 minus theta right so just you assume one angle is Theta and you can now solve everything right so you can see that now these two triangles are similar by a similarity because all these three and you&#39;ll see a synonym is what if two angles are equal obviously third one has to be equal because it is it is what it is sum has to be 180 degree okay so here also all three angles are equal that means with that means what attitudes triangles are similar so abd is similar to triangle BDC correct so if I write the ratio you write write the ratio you how do I D ratio now you can write D ratio you quickly again by picking angle right you will decide which triangle to keep in numerator so let&#39;s say angle a triangle a we need to keep in numerator and triangle BDC to keep in denominator right okay so now we can solve now in numerator triangle a way to pick any end and suppose you pick theta at first so here theta opposite is ad here theta opposite is ad and here theta poet is what BD so I can write the ratio as AD upon BD okay next one you get picnic angle is 90 suppose right so here in triangle abd 90 opposite is AV and in triangle BDC 90 upon it is BC so KB upon VC try it this is how you write the ratio in every question based on the angles right when you prove by a similarity you must have mark the angles right so use that mom tengas for adding the ratio right no other method simply didn&#39;t directly do it right third one let&#39;s say let&#39;s say third one I bigger is 90 minus theta so here 90 minus theta opposite is BD here 90 minus theta pulled is what DC so my ratio is what B D upon DC fine so I can combine these two and from a result result we&#39;ll discuss later also so you simply write here BD square is equal to 80 into DC right so like this you can have any kind of any or any pair of triangle you can take right this is the first pair I&#39;ve taken now second pair is also similar right that means have taken fin first one I have taken two smaller triangles this one&#39;s one right now second one I can also take the bigger triangle triangle ABC is also similar to triangle BDC again by this is equation 2 I can write a similarity and again I can write D ratio right same way because for both these triangles be bigger and is monitoring this all the angles are what equal right now again sudden third also generate the bigger one triangle ABC is similar to a smaller one again triangle ADB angle ADB right so this is a very important case of right angle triangle inside a right angle triangle right will again discuss it later right we&#39;ll come back to it okay so to find all the values right so this three are very important of similarity because when you write a right angle triangle in a right angle triangle all the three pairs of trainers are similar just two smaller one one bigger one and one lower is smaller one one bigger one and one upper smaller one right all three pairs are similar right and so by similarity can you remember is it also yes right and remember is that here you will come back to it but it&#39;s okay more you revise it more you will learn quickly right to find sides in triangle right now let&#39;s say this is a this is B and this whole is C this hole is see so it&#39;s a result of similarity I am writing here it&#39;s a result of similarity this hole is see right the result is actually if if this is a this is B this side is this side this is smaller one I could write a b c and d this ad is nothing but a square by C this C D is nothing but B Square by C and this B D is nothing but a B by C okay is reverse result as a result of similarity only again all these three sites three purifying triangles will put similar handle get this result very point is out right indirectly right this is a this is B this is C this is very easy to remember also this is a square by C this is right ad and red double ad ad s ad is a square by c BD is sorry C D is v square by C so you write here a you right here bu right here C upper one is ad is a square by C lower one BD BD is B square by C and so it not be d CD CDs B square by C and his BD is what this PD is a B by C right so very quickly can get this value for example we have questions also like this right suppose is a triangle here and triangle is sides are 6 8 and 10 ABC 6 8 and 10 and this BD is the in height of the triangle right so what is the value of ad what is the value of D C what is the value of BD right so in one go you can read in one go you can write write what is ad what is the ad here ad is basically a square by C what is a square six square so here it is a t-bird 6 square by 10 that is 3.6 this is ad so ad is 3.6 right what is C de Sade&#39;s B square by C what is B Square basis is B right so 8 is square by 10 that is 6 point 4 it says this is BD so this is CD CD but is BD a bit is a B by C but if I give you I see 6 into 8 by 10 that is 4.8 therefore I can write BD is what we D is 4.8 right so not at all time taking simply you write this value get this answer anytime okay it&#39;s a very good concept remember it okay by similarity and again we&#39;ll come back to similar active adding a triangle again okay so for now what I can write here is okay this to a question you know just do a question here and the question is there is a there is a a square inside a triangle okay liquid rotor angle here there is a square inside a triangle like this okay this a square inside a triangle okay and sides of a triangle are basically twenty thirty four and forty two this is 42 this is 20 and this is thirty-four right this whole side a b c fine find a the question is find a length of side of a square side of a square right again a super cushion which can be solved by similarity in a very easy way right if I understand this right it is a super application of similarity see now when you have figure like this here so I know they did not get all the hidden facts sort of the figure right it all the hidden Friedan facts out of the figure now since it is a square so square means what opposite sides are parallel so this is parallel to this obviously now name it de we can name it de here so now de is parallel to BC so again same thing this will be corresponding angle and this will also be corresponding angle that means I can write that triangle ABC is similar to triangle we see okay so it is similar so I can again write in our issue okay write the ratio how to read e ratio here so now angle a is the comma and the third angle right so you see if you pick mark this as theta you mark this theta theta so modulus Phi and Phi right you see here I home NT side of a square right 12 side of the square ladies side of a square ba later this side of a square by a so all four sides will be equal to the square means all four sides equal right so I want a so I should pick which and then I should not pick theta here because theta opposite here is Haidee and it is not known and neither I am interested also write Phi opposite here is a again AE is not known right what I&#39;m interested in is I need to find the side of a square right so I should pick what I should pick angle a here here so that is a common angle in both a triangle right so here the common angle opposite is a in smaller triangle in bigger triangle common angle a poet is what 42 so I should write a upon 42 okay should write a upon 42 okay no you should also define your red cape on for it equal to I I need to find a value of a right now equal to what equal to what right so this is equal to what right so this is again I need four other angle right so again I told you if I pick the common angle it is fine but again if I take if I pick fire posit fire point a is not known already a is one unknown variable another one unknown AE will come or if I pick theta another unknown ad will come so that means I can&#39;t pick so ratio from ratio of corresponding sides right so remember the statement when two triangles are similar ratio of corresponding sides is equal to ratio of height and it told you earlier when all three sides of a triangle are given I can get the reach I can get the value of height right now by equating area by equating area right so same thing I should do here also okay let me rub this part so basically I should write to the ratio of heights I should write to the ratio of heights right now to rate ratio of height here so ratio height can be written let me draw a height here draw I tried this height for both the triangles fine a F and a G okay how did you how did at the height of triangles see height of triangle you can get by two method first one is by area okay first one is by area okay see so what is the area of triangle root under s into s minus a into S minus B into s minus C s minus B into s minus C when necessary perimeter right what is area here so area is what is s here s is 20 plus 34 54 plus 42 96 s s 96 by to pretend this is what I do 48 so 96 by 2 is 48 so s is 48 here correct s is 48 here ok now 48 into S minus a 48 into s minus a right what is s why I say 48 minus 4 T 2 is 6 now s minus B 48 minus 34 is 14 s minus C 48 minus 20 is 128 now what is the area so area a little bit by now see you have got two multiples of 7 year right that means we have got see what I can do here I can split this 6 as 2 into 3 so this 2 with 14 will become 28 so out of 228 128 will come out right out of 228 128 will come out ok what is left here now so 3 is left so 3 into 48 is what 144 3 into 48 is 144 and what is square root of 144 12 so area is what area is 28 into 12 that is 280 plus 56 that is 333 36 this is the area of trundle ok and this I should equate with area of triangle is equal to 1/2 into height into base right what is area 3 3 6 what is height here you need to calculate what is base here for height and base here 42 right so you get the value of a solidity value here simply cancel eight times right so height you will get as 16 height you will get a density right so by this we can get hide as 16 okay now this is the first treasure to get the height right obviously should do the concept right no there&#39;s a shortcut also and the super shortcut right if you start using Pythagorean triplets now you will find gym it really easy right because 30 to 45 no question you enjoy solving because of Python Spicer interpret right so in second method I&#39;ll use PI 0 temperate right remember that I heard I got height as 16 okay I got high death 16 fine height as 15 I got okay now by use of second method by use of Pythagorean triplet okay so how to use the power to use Python interpreter see method to five Pythagorean triplet see the triangle here triangle is 20 34 and 42 I am calculating height of the triangle try it now you just get a triplet we just did a triplet triplet with hypotenuse 34 right so just cross-check all the triplet right because 34 can be in any triplet also or it can be derived right can it can be multiple of an interpolated so third the as soon as you say 34 it strikes that it is a double of 17 and double of 17 means what the relative order plate is triplet is what triplet is eight to fifteen seventeen okay how again 34 I&#39;d look at 34-year again you cross check those six basic I told her to remove those six basic triplet right 3 4 5 5 12 13 7 24 25 and 850 79 40 41 right so all those you cross check here and you see that 34 34 is a hypotenuse it is a multiple about 17 then it all the hypotenuse you write 3 4 5 5 is the hypotenuse 34 is not a multiple of 5 5 12 13 again 30 is the hypotenuse 34 not equal to 13 now similarly 24 25 25 the hypotenuse 34 is not a multiple of 25 then 8 15 17 right 34 is a multiple of 72 times right so if a triplet is 8 15 17 and it&#39;s double of it it develop it right so should I should write okay double is what 8 15 17 is 34 what a double of it never is 16 30 and 34 double is 16 30 and 34 right now I can&#39;t write I can&#39;t write this side as 30 why because in a triangle this is a triangle this is a triangular in a triangle hypotenuse is the longest side okay and 20 is the hypotenuse it is a longest side right so in this side can&#39;t be 30 so 16 and 30 so she should be 113 should be this one and it should be 16 okay so this is a fundamental iterate again make it a little bit of complications in the diagram where I did let me make it again okay just concentrate a very good concept here to save you time 20 34 42 this is the height so it should be what it should be 8 15 17 devil I told you that is 16 30 and 34 now not always too right so we know that this is this side also it should satisfy because this is also right angle triangle so it should satisfy this because I have made a bit of guesswork right so just check cross shape is satisfying or not it is satisfying C is 42 this 30 so it should be - all right this is writing with triangles they said also adding with language at 12 16 20 and under triplet right derived from 3 4 5 3 4 5 into 5 times okay so I 3 4 5 into 4 times 2x sufficient when this is at satisfying right this satisfying what is the height so height is 16 so height we got a 16 right stretch try see you can always do it this but it is a concept right but try this kind of logic right you should try this kind of logic and you can read the quotient within minutes right this these things are very important very very important right so try using these things okay triplets triplets engage a right angle triangle is a super concept right you should always pray by putting triplet try to recognize it or play it right you can save a lot of time here okay so anyway I got height as 16 right I got height as 16 okay now come back to this question having all this right now height is 16 okay so that means what basically now the ratio sites of toot - triangle right so common angle ability is a common angle opposite is 21 right common angle a poet is a common angle suppose it is nothing but 21 okay this is a height of triangular so 21 right now no so ratio of two&#39;s ratio of sides of triangle is equal ratio of height ratio of idea so then you see this is a this is a right this is a that means it&#39;s a straight line is a square is ninety degree three also 90 degree so straight line that this also should be a this also should be a so this is a hole is 16 so high it off a smaller triangle how much 16 - a pride 16 this length is a this length is a this FG is a right FG is a why because parallel to the side of a square okay so a holy 16 so height of a smaller triangle is 15 minus a upon height of bigger triangle that is 16 right so this is the ratio here we got here it&#39;s a super question letting get the value of a from here 16 a is equal to 42 into 16 right 420 + 252 that is 672 672 - 42 42 K right that means 58 a is equal to 672 therefore a is equal to 672 by 58 okay that is 336 by 29 so answer is what 336 by 29 this is the answer for this question what is the value of a 336 by 29 okay so very nice question lots of concerts record is this video right especially this question I understand this question you can apply in many concept many other question right this concept okay thank you we&#39;ll continue next video [Music]

Transcript for:Geometry Class: Similarity of Triangles

Transcript for:
Geometry Class: Similarity of Triangles