[Music] [Music] hi everyone my name is Ravi Prakash and this is his 11th class of geometry okay so we are continuing this can concept from tenth class so geometry only right that is previous class now the question here is how many distinct triangles can be formed with integral sites if perimeter of triangle is equal to 39 okay and with all sides odd and with two sites even at one side all right the similar question we did in the law at the last of the previous video where we discussed about the same question where the perimeter it was even number here the pewter is odd number right so how to tackle this kind of question we will see how see with perimeter with all side three sides as all right so let's say again Omega triangle ABC sites are ABC right it is given that a plus B plus C is equal to 39 now have tobik all sides all right how to make all sides all so I'll replace a with to a minus 1 I will replace B with 2b minus 1 and I'll replace C with to C minus one wicked capital a capital B capital C is equal to 39 right now why have taken to a minus 1 because I want to start from one right no not to a plus one because they want to start from I want to keep a equal to zero so ad sorry I want to keep be equal to 1 so at a equal to 1 at capital a equal to 1 is small a will be equal to 1 right and whatever value you you put for this capital a now is small hey will always be order starting from 1 right so that's how you make any number s all right so to a minus 1 plus 2 B minus 1 to 2 C minus 1 equal to 39 ok now so to a ok plus 2 B plus 2 C is equal to 42 therefore a plus B plus C is equal to how much 21 okay so a plus B plus C is 21 now now normal solution comes to rate is 21 it's an odd number so total number of triangles for odd number how much so total number of triangles for odd number is how much P plus 3 square wave 48 right remember P plus 3 whole square 48 where 48 is in here inside in here it is a function it becomes 24 square by 48 right so it basically becomes 24 square by 48 fine that is again same thing here 24 square is 576 by 40 into 2 12 so 12 is the answer for this question right so how many triangles with all sides or there will be 12 triangles I can form it all side spin all right okay fine now this is for this answer first portrayed is answer first part okay now second part again connect to previous video last question right second part I'll discuss them let me rub this part so second part was basically if you remember it consisted of only two pieces right in the aft case case it was given number even number it consisted of only two cases because in first case always even even even second case two was all in one was even photos already given number right these are talking about last video last question in this question now you see again should right solution for two solution for two again no a plus B plus C is equal to 39 great right but a plus B plus C equal to 39 now and here this understandest concept here right allocate odd result is 39 is odd when L get odd I'll get odd only when a is odd as well as B is odd as well as C is ordered right so odd plus odd plus odd will give me odd this is a first case this is the case one right and another case I'll get all result as odd as any two as odd so sorry I need to one as odd and 2's given right that is even plus even plus odd will also give me all right is there any other case with a plus B plus C equal to 39 that is an odd number so with is is there any other case apart from these two where a plus B plus C is an odd number only two did no other case apart from these two first case odd odd odd second case even even ordered right okay now what an answer for first case we remember last question right answer worked well we just solved it answer was too well for this question right now how many total number of triangles what is the answer for this case now you try it so answer for this total Rangaswamy total triangles two total triangles here is now what is total diagram is 39 is an odd number right so P plus 3 whole squared in 48 so 42 is square by 48 this is the total number of triangle right what is 42 square 1764 by 48 in nearest integer function okay now how much it is so we can just solve it 48 into 3 is 172 32 + 4 324 so 48 into 6 is 288 around it is 36.7 hour roughly right that way it is more than to this point five right so if it is more than thirty six point five this have to with 37 triangles there have to be 37 primaries right very important so there are seven 37 triangles here so total trang this is 37 out of this case one per angles case one Prangley how much 12 right so obviously 30 sign will be Ward sum of 12 plus this one so what dish what is what what red of death 25 so 37 minus 12 what is answer 25 so 25 is answer this question right there for question 2 what is answer answer for question 2 is how much 25 right so very good question again same thing right a plus B plus C is an odd number it will be odd in only two cases when all three are odd or when any two are even and one is odd right so even because even is even plus even is even and odd given prasad is odd so total unit is all right so this total number of triangles consists of only two cases right either case one or case to sort not either case one case one and case two so I hope you already solved for case one the previous question it was it was 12 once I was 12 right so out of 37 triangles pulled out of total 37 triangles to values in case one so how much is it for cage 225 21 sir so 25 is the answer right very good question very good question please please put a star mark and revise this question right okay now next one so this this kind of concerts we have covered right every confidence is canyon be covered I've covered all the concept condition for triangle formation acute obtuse right angled triangle is Callen is LS equilateral total total level triangles when parameter is given also right so every case is covered not a single question can be asked in this concept in your inning examination my guarantee right now now different question you take a different variety a question if perimeter of a triangle is equal to 60 then then which of the following which of the following cannot be d cannot be the area of the triangle area of the prandtl right put some options here 151 be 159 see 169 and D 176 okay so if perimeter of a triangle is 60 then which of the following cannot be the area of triangle right and we can write one important concept here if perimeter is constant if perimeter is some constant then area is maximum in case of in case of equilateral triangle in case of equilateral triangle right so you get a maximum area in case of equilateral triangle right so what is triangle so triangle is 8 2020 and then the perimeter is basically an equilateral time what is the area of a political equilateral triangle so a area of equilateral triangle remember for now we'll discuss it at a tangent detail right area for equal equal at a times how much root 3 by 4 into raised square into side is correct but William will test chamber for now we will discuss equilateral triangle in detail right in the coming videos right so root 3 by 4 into side is square so root 3 by 4 into 20 square right that is 100 root 3 root 3 is around 1.7 3 so 100 into 1 point 7 3 so you see it's around its around 173 so 173 can be the maximum area can be the maximum area right so which of the following cannot be the area of the triangle which of the following cannot be the area of the triangle so obviously if 173 is maximum so ABC can be the area of triangle D cannot be the area of triangle right so answer for this question is d ok because pyramid perimeter is constant so area will be maximum only in case of equilateral triangle right that means when that perimeter is equally divided into three sides so if a is 16 20 20 and 25 so what is the maximum area I'm getting well if perimeter is 16 the maximum area I can get is 173 right but option contains what 176 and question was question was which of the following cannot be the area of triangle so it is answer 176 ok so again a good concept right wrote it down now next one we will do next question rift concept we'll do this next concept see in triangles there are two important concepts like sign rule and cosine sine rule and cosine rule right now let me make a triangle here I'll make a triangle ABC triangle a BC by convention how we write capital here put it a small a capital B avoid a small B participate a small generator sites are a B and C right now this is angle a angle B angle C so what the sine rule say sine rule says that a upon sine a is equal to B upon sine b is equal to c upon sine c right if you don't remember the values of sine and cos I'd is a basic geometric we should remember it so please watch my video basics of trigonometry I have discussed about derivative basics they write so a by sine EB by sine leave C by sine C this is the cosine rule right that means side and the sine of the opposite angle so a upon sine a right against side and this sign of the opposite angle b upon sine b side that is c upon sine of theta polar angle that is and i'll see write this sine rule okay now cosine rule now cosine whose root is very important cosine would is very important right see how to read cosine will do so if I read cosine rule so what is cos a course of angle a so cos a is C or I remember it actually you have to write sum of squares of all three sides but opposite will go in negative that means B square plus C square opposite will go in negative whichever angle you chose in radial taken cause cause a so it's this length B square plus C square and opposite in negative that is minus a square minus a square upon upon twice of toys off for a product of two sides but don't take the opposite side that is twice of BC so twice of Missy right this is cosine rule right how to remember simply sum of squares of all three sides but opposite one will go in negative so B square plus C square minus a square upon twice of product of two sides and don't take the opposite one okay this is cosine will four cause angle a right I can I can oh sorry to a cost cause be your site for Cosby Cosby what is the cost be so Cosby Apple it is a small B right that means I can write a square plus C is - B squared sum of squares of all three sides but opposite cosy negative so a square is C square minus v square upon twice of product of two sides don't take opposite one what is the point earlier the SIRT is twice of AC correct okay we can also write ocassi also becomes red for cause see here right so cause see here will be how much you can write by yourself now what is cause see again sum of squares of all three side what is cause the cause of an this see the sum of squares of all three sides up one goes a negative so a square plus B square minus C square upon twice of AC sorry choice of EB don't take therefore you do it twice or maybe write this cosine will is super important can apply it many many situations in geometry right and there because it involves one angle what is cosine will do it involves it involves one angle and three sides so total four things it involves one angles and decided three sides right so when three sides are known you can calculate the angle by cosine rule or when one angle and two sides are known you can calculate the third side by cosine row right so out of four things whenever three things are unknown three are known sorry three are known so one unknown can be easily find one unknown can be easily find right this is cosine rule right very important cosine rule okay now let's do a question you type a question the triangle PQR okay p.m. is the angle bisector that means this angle and this angle will be equal so p.m. is the angle bisector okay angle Q plus angle R is equal to 120 degree given PQ is 9 PR is 12 okay PQ is 9 PR is 12 so question 1 question 1 what is the length of QR question 2 what is the length of p.m. two questions here right so a simple application of cosine rule I will solve it so that so that you get the concept here okay see first we can find a length of QR right QR you can easily find because and then Q plus angle R is equal to 120 degree that means sum of three angles in a triangle is 180 degree that means angle P angle P is equal to 60 degree so when angle P is 60 degree and PM is the angle bisector given PM is the angle bisector that means this is 30 this is 30 correct and this whole is 60 this whole is 60 right that means again two sides and one angle are known I told you in cosine rule there are four unknowns for four things here out of the three things are known two sides and one angle is known so I can easily the third side that is Q R right so I can apply for cos cosine will again apply for cos 60 so cos 60 degree that is cos of angle p.m. applying what is cos 60 now again sum of squares of all three sides but opposite will go in negative that is nine square plus 12 u square minus Q R square upon twice of product of two sides don't take the opposite one that is twice of nine into 12 a point is Q I don't take the opposite event right what is cos 60 cos 60 is half and so on first question here cos 60 is half if cos 60 is half so this is how much 81 plus 144 225 minus QR square by this is two one six okay so this is this will be 1 0 here so this will be like QR square is equal to root 2 1 sorry not root QR square is equal to 1 1 7 that is 13 into 9 right therefore therefore Q R is equal to 3 root 13 this is the answer right so QR a got the first question what is answer 3 root 13 4 first question you got the answer I got the answer right now second question second question finding the value of PMF right so finding the value of field now see they said there will be main thing is tracking in your mind how to find the value of p.m. right let's apply that triangle area formula because all three sides are known and I can get the value of p.m. right or we can apply angle bisector theorem also what is angle as it sort of so when p.m. is the perpendicular bisector sorry angle bisector then the foot of the angle bisector by set aside in the same ratio has the other two sides of the triangle that means what is the ratio here 9 if to 12 that is 3 to 4 this also will be 3 to 4 right and you can divide total loaded total in here 3 root 13 so you can divide 3 to 30 in the ratio of 3 to 4 right but you see it very complex right very complex them because it's already in these roots huh and when all three sides are known and you have Leonard formula supply that formula right before that term because it does that what is the form of form of idea of triangle root under s into S minus a so already there is one root another hood will comprise so solving is very complex ID so that's why that 5th way of calculating area of triangle never forget that it's a very good formula right fifth way of calculating area of triangle is what area is equal to half a B sine theta of a B sine theta and I've discussed his concept earlier also that if total area is given that in distributive eager total area is equal to sum of parts shouldn't some two questions on this concept read total area divided into sum of parts right that means area of triangle that basically means what that basically means area of triangle PQR okay area of triangle PQR is equal to area of triangle PQ m plus area of triangle PM r k of trundle PMR right but is a do triangle PQR so area of triangle PQR is how much half a V sine theta so half into 9 into 12 sine 60 Family Safety right so half into 9 into 12 sine 60 degree is equal to area of triangle PQ what is a rotor angle pqm so PQ m area is how much half into nine half into 9 into PM sine 30 angle between adjacent 30-degree right so two sets are two sides have you gather together it'd be Muslim so half into 9 into PM sine 30 degree plus area of triangle PMR right that is half into PM into 12 sine 30 degree right so you can solve this will get the answer you can solve this will get the answer for p.m. as if Abby will yard 36 root three by seven can solve it a check so answer for p.m. is 36 root three by seven right so good application of cosine will hear a good application of course I'm Maria okay okay thank you will read it in this video [Music] [Music]