[Music] [Music] hi everyone my name is Ravi Prakash and welcome to the third class of geometry okay so in geometry 2 we discussed about all the four gentle centers in center circumcenter also Center centroid and all the points which relates to those four centers right now here we will discuss some very important points related to triangles okay like some properties of medians or operons theorem angle bisector theorem right so very important points will discuss a right which many of you must not be knowing also will discuss those points in right okay first point so forth theorem is normally we'll discuss Apollonius theorem of Longinus theorem okay I don't know name is how much important right Apollonius name right but it's application okay so Apollonius theorem the name you may not remember but you should remember its application okay so it's application is what it is related to medians very point point reg which theorem to be used with okay so related to the median so whenever I have to find the length of medians or I relate something relate something that is related related to length of the median side then I'll use a pronestyl pride so now it's easy to remember also will write Apple are sort of the remember it okay Apollonius theorem note is it C ABC a trundle since it is related to medians selected for all the three medians let's say it is a median for the median 9 joini midpoint so D is the midpoint of BC similarly e is the midpoint of AC and similarly F is the midpoint of Av CF is the median midpoint miss this f be equal to F a fine so the theorem here is if you took one time if you take ad as a median right if you take ad as the median if you take ad as the median right so this theorem is twice of a d square plus right that is the middle square okay twice of ay d square at the middle square plus C D or D be right that means on whichever side it is that media is falling right so since D is a midpoint so DC and D B both are equal so I can take either of this right so ad square plus C D square or DV square let me write C D square ad square plus so ad square plus DD square or I get a DV square since both are equal is equal to is equal to surf some of the lengths of squares of the other two sides right that is right now ad and BC are involved a B and AC are not involved right so this is equal to a B square plus HC square so you can remember this very important theorem okay twice whichever median you take right twice of that video is middle square plus on whichever side it is falling that is square right so that means CD on which was video said it is falling it will be in two halves right so take either of the F so middle square plus either of the half is square it is let us say CD square here is equal to sum of these squares of the other two sides which are not involved right now that is a B and a C okay let me read one if I take B II as a medium if I take B II as a medium so what what can be by theorem my theorem will be twice of first you write twice median squared is B square plus on whichever side it is following that either half I need to take right that is AE and EC both are equal plus AE square okay is equal to sum of the other two site that is EB and BC now Evie square plus BC square so very important a branch theorem you can also write when CF is the median so when CF is the median so this theorem will become twice of CF is square plus F a square or FB square is equal to AC square plus BC square AC square plus BC square so this is a Piranha's theorem make is an equation 1 this as equation 2 and this as equation 3 okay so now I'll write these three conclusions of operons theorem right and it shall write three conclusions of Apollonius theorem that will be that we'll get by adjusting this one two and three that means by adding or some adjustment we'll get this three applications right three congresses okay so conclusions on apollonia surrounded slide first conclusion is I should write conclusion of on trousers on Apollonius theorem Apollonius theorem okay so let me write it triangle here make it liquid with the triangle here ABC okay in this ad is the median B II is the median and CF is the median okay and let's say G is the centroid okay G is the centroid right now first conclusion first conclusion if you add all these three equations there if you add all the three equation editor one two and three the result you will get is thrice of a B square plus BC square plus C square is equal to four times of a d square plus B square plus C F square so very important point first one if you add those three and make bit of adjustment there's the first complemented thrice of a B square plus BC that is all these sides a B square plus BC square plus C square so I can write like this or a ad squared plus B squared plus C F square upon upon a B square plus BC square plus C a square okay what is ratio so ad by CV what is is 3 by 4 this is 3 by 4 very important point okay very point point 5 now second point again which bit bit of adjustment is get dis AG square okay AG square plus BG square plus CE G square upon right at is a B square plus BC square plus CA square is equal to 1 by 3 is equal to 1 by 3 right again every point point so this is AG right what is gay G with the median right so if ad is the median a gh2 GD is in the ratio of what 2 is to 1 because centroid is Android right the centroid divides median in the ratio of 2 H 2 1 okay so AJ squared plus B G square procedure square upon EB square 2 BC square by C square is what is what is equal to 1 by 3 okay is equal to 1 by 3 this radiated by just breaking it right in writing you can write e d square here s hg + GD and then you can get this result here right fine ok now third a point in a piranha serum right third point so third conclusion of Apollonius theorem third component theorem okay the third conclusion of a pelorus theorem is anytime right it is any time when you write these sum of pay meter right whatever is the perimeter of the triangle that liquid idea that whatever at the perimeter of the triangle so perimeter of the triangle will always be will always be less than four by three into some of the medians okay and it would be always greater than the sum of the medians some of the medians okay so perimeter of the triangle will always be less than four by three into some of the medians and it will always be greater than some of the medians right so again a super important point this is for the perimeter of the triangle okay payment of the triangle so these are the three conclusions of Apollonius theorem right we are still the first point we were to viewer discussing very important points in transits right after this we'll do questions once we discuss every point point in triangles right and in theory which will continue obviously right but still very point points let me discuss here that Apple on a serum and it's 3/4 presence of a polyester right now second point in into second point second conclusion is a second very important point in triangle is but it relates to basically it relates to finding area of the triangle when lengths of the medians are given right so when lengths of the medians are given how to find the area of the triangles right so now see what is the area of trundle eg of triangle unit by here I'll tell you the four five ways of getting area of triangle I data okay that you can get but here how to mention those points in application to length of medians right so right very point point here if if lengths of medians are given if lengths of medians are given then then area of triangle sorry then area of triangle then area of triangle formed by formed by length of the medians length of the medians is is 3 by 4 of the area of original triangle right again a super important point rate if length of medians are given their area of triangle formed by length of the medians is three by four of the original triangle now what is mean you write it is mean here that if for any triangle C what is media it for any triangles let's say triangle ABC is no there and again hi know the medians ad b e and c okay so if the length of the media it is given that DD let's say KD equal to nine be e equal to 12 and CF is equal to 15 it is given for menial intermediates are given right ad is nine B is 12 and CF is 15 so I've been hashed we had a question here this will get question right so what is the area of area of triangle ABC what is the area of triangle ABC right so the point here is when the length of the medians are given so if we try and find out a imaginary triangle with the length of the median sides if you if you make an imaginary angle with little medians so let's say I made a triangle here this with length of the medians here so it lends Albany medians are about nine to eleven fifteen so one side nine one side twelve and one side 15 right so whatever the area of this triangle at this area Indian because this is this is right angle triangle because this is this is satisfying Pythagoras theorem right this is in the next slide we'll discuss how to find the area of triangles and all right this is for this slide so itit simply satisfying by theorem 15 a square equal to 10 is Kurdish 12 is square that means it is 90 degree but if it is 90 degree that means this is a right angle triangle right so it is area of retinal triangle area is half into length half into product of the two sides except hypotenuse right at is half into we leave the hypotenuse right so half it to lengthen the breath right so this is this could be length this could be breath right and this is the hypotenuse don't take hypotenuse half it will end up with so half into 9 into 12 okay so half into 9 into 12 this is what this is how much it is 54 right so area of triangle ADF and hypothetical triangle formed by the length of - is 154 is an hypothetical triangle and imaginary dragon right that that area is 54 if I take the length of my dear as a side of the triangle that area times different people this point says that this 54 is equal to 3 by 4 of the area of original triangle right this is equal to 3 by 4 of the area of original triangle right that means ei of original triangle is word 54 into 4 by 3 since almost 70 to 72 this is answer ok so again a very important point if lengths of unions are given then the area of triangle formed by the length of the media's that area of that I get I can write here also area of imaginary triangle formed by the length of the medians is 3 by 4 of the area of original triangle okay so this 9 12 15 and sides of triangle the satisfying Pythagoras theorem because larger side square is equal to sum of the squares of the other two sides it is PI theorem we'll discuss in the coming slides okay it was the area of triangle so if it is writing in dry against the edge of tangle tangle always water always half into product of the two sides except hypotenuse right that is half into 9 into 12 okay not hypotenuse at a 50 not hypotenuse right it is on cell 54 and 54 is 3 by 4 of the area of voltar and then okay so this is a second very important point in finding length of the millions okay now third very buoyant point okay third very high point point right itself right it relates to the distance between circumcenter and in center it relates to the distance between the circumcenter and in center okay right at third point now distance between distance between circumcenter and in center okay this is given by this even by d square is equal to R into R minus 2r ok d square is equal to any point formula right it is remember it is formula very point formula distance between circumcenter and in center of any travel is given by would be square these are distance right this is the datas is a circumcenter see is the in-center I this is a D distance right any triangle centers water inside any translating talking about like this so okay so me the tranquil you make of it a diagram okay so this is like circumcenter is like in center talking about inserted randon these two points realize have some not always head we'll discuss when we like inside when layers outside right this distance is d the distance is olga invoice circumradius R capital R is a circle it is recall is rainbow capital R is the circumradius and a small R is the in radius so R into R minus 2 1 ok fine now see now since it is a d square here so D Square will always be what we'll always be positive right that means I'll always be greater than equal to 0 and and always bigger than equal to zero it can never be negative square of any number can never be negative right that means R into R minus 2r will also be always greater than equal to zero that means R minus 2r will be getting equal to zero that means R will be critten equal to tour or that means that basically means that r by r is greater than equal to 2 that means ratio of circumradius and in radius in any triangle is always what is always greater than equal to 2 a very point point and is only a and is equal to 2 it is equal to 2 only in case of equilateral triangle that we'll discuss further right so this is this is the sign contains greater than or equal to so equal to is only for equilateral triangle for all the other times it is greater than 2 okay greater than 2 is again a very important point distance between circumcenter and in center ok now fourth important point four point one point BM is how to find the area of triangle how to find the area of triangle right so area of triangle can be found in five ways basically typically five ways and that will be covered in all geometry okay so let me write all the fibers here itself first one is okay that first one is area is equal to root under s into s minus a into S minus B into s minus C where s is equal to a plus B plus C upon to write is a triangle containing sites a B and C so it is s s s semi-perimeter so what is perimeter length of whole boundary that is a plus B plus C say semi means what half so say what is same way better half of the perimeter that is a plus B plus C by 2 this area of triangle right first way second way here a is equal to always half into base into height right so whenever a height of triangle is given what is the area half into base into height right that means patrol triangle like this a PC so here a B is the height okay what is the area of the triangle so area of this triangle will be either of the SRAM will be word half into ad into BC half into height into base right see this height and base are corresponding terms right because the in triangle entangle there are three Heights there will be three Heights right if I draw the height like this if ad is the height for me so in that if a DC ad in that case BC is the base but I can also draw a height as C so if C E is the height for me then a B is the base similarly if ve is the height for me then AC is the base right so height and base are corresponding terms right on whichever side height is falling that is the base okay so the base here is what the base here is BC okay second way now the third way is area is equal to basically R into S right is our we've already discussed but is our his R is basically in radius this R is what in radius okay now fourth one is what fourth one is area is equal to kV C upon for our right is for is capital R you know what is capital R we already did already discussed what is capital R circum radius okay so ABC upon for our this is the capital R eight a equal to ABC upon four okay so four ways and the fifth ways the fifth way is area is equal to half into a into b sine-theta right what is theta here theta is the included angle theta is the included angle between between a and we ride that means whenever I have two sites given let's say this side is length a this side is length P so if theta is this angle then I cannot calculate the area of the triangle because theta is not the included angle what is included angle that means including those site right put asides here a and B so in this case I theta is not be included and written a and B so you cannot get the area here I can do it right here I can get if I made a triangle like this if this side is a this B since theta so here a a I can get an area is what what half a B sine theta by very point half a B sine theta okay super import formula fifth one right generally forward in the heat of those four formulas are more frequently use right but this is also very important of a B sine theta so see it's any sided it is very easy also suppose any triangles we have been given and in two sites let us give an are three and four okay suppose a triangle is given here this side is three this side is four it is 30 degree right is very easy getting area of the triangle areas word half a B sine theta so half into 3 into 4 into sine 30 degree okay so half into so sorry it is how much it is 6 into what is sine 30 sine 30 is 1/2 so a day is what area is three three units three square units is the area of this triangle fine so sine 30 is 1/2 right so the sine 30 and all have discussed in the video called basics of trigonometry right please watch that video so we discuss about all the concepts of derivative okay basis or trigonometry and all its application have written there so so please watch that video ok fine so this was like area of triangle okay so initial i'ts will discuss in for few important points angle bisector theorem and all and then we will move to some good questions okay it's their applications whatever is reading right now okay thank you [Music]