 Okay, good afternoon everyone. Is my voice clear? Okay, can I just want one thing? NPSRNR, any one of you kindly send me the syllabus for October 4th exam and NPSHSR also can you please, any one of you can you please send me the syllabus for your exam and the date also. Okay, so in the meantime you guys send me the syllabus. I am starting with number system because number system is part of Niharit saying it's 3rd October. So this is 3rd October. I am writing down the syllabus which I know for most of you that is number system then you have polynomials and you have guys can anyone send any other thing number system polynomial is algebraic expression part of syllabus if you guys are not sending it. Okay, so Shreya is saying they have in NPSHSR the syllabus is number system polynomials, coordinate geometry, eucreate lines, angles, triangles and herons, heros formula. So for NPSHSR guys, I have already covered angles triangles you can look at other two videos at Centrum Academy site and heros formula in the first lecture. HSR number system, okay so nearly heros formula is coming so probably I will cover heros formula once again, polynomials number system. Okay, so let me start with one question of number system which is based on rationalization. So I am starting with a very easy question. So it has been given that 4 plus 3 root 5 divided by 4 minus 3 root 5 and you have to convert it in this ratio a plus in this expression sorry, so a plus b root 5 this question is nothing but to tell you or this question is expecting from you to do what to do rationalization of this. So what I do is that somehow because the denominator on this side is 1 I need to make denominator here 1. How do I make denominator here 1? So I convert this negative sign to positive sign and multiply it in both numerator and denominator. So this is nothing but 4 plus 3 root 5 divided by 4 minus 3 root 5. I expect all of you to keep on solving questions with me and keep on giving writing answers in the chat box 4 plus 3 root 5 divided by 4 plus 3 root 5. So what do I do? This is nothing but a plus b whole square so a square is 4 square which is 16, b square is 3 into 3, 9, 9 into 5 is 45, 3 root 5 square is 45 and 2 ab is 2 into 4 into 3 is 24 root 5 divided by 16 minus 45. So how much I should write this? I should write this as 61 plus 24 root 5 divided by minus 29. So if I have to convert it into a root a plus b root 5 I will write like this here. I am writing it here. I am writing it minus 61 by 29 minus 24 by 29 root 5. So now start comparing a would be minus 61 by 29 and b would be minus 24 by 29. Now let me move to other question. So one good question is this and then a few of you have also send me the doubts which I will be taking at the last last 15-20 minutes I shall be taking doubts. So question is this is equal to 3 plus root x. So find the value of x square plus 1 by x square. So if x is 3 plus root 8 what I need to find out? I need to find out what is 1 by x. So this question is also about rationalization. How do you rationalize it? You write 1 by x, 1 by x would be 3 plus root 8 and try to eliminate denominator from here. Eliminate denominator by writing 3 minus root 8 and 3 minus root 8 here. So that would be nothing but 3 minus root 8 divided by 9 minus 8 which is equal to 3 minus root 8 divided by 1. So 1 by x is equal to 3 minus root 8 x is equal to 3 plus root 8. Now what I need to do? So try to understand this can here it is not x square minus 1 by x square that you will write something like x. So the best way to do is that x plus 1 by x whole square is equal to x square plus 1 by x square plus 2. Now if I have to find out value of only this much so I will bring 2 on this side. So x square plus 1 by x square can be written as x plus 1 by x whole square minus 2. Now x is this much and x so let me find out the value x plus 1 by x would be equal to x is 3 plus root 8 and 1 by x is 3 minus root 8. So what happens this root 8 root 8 is gone 3 plus 3 is how much 6. So I put this value 6 square minus 2 which is equal to 36 minus 2 that is equal to 34. So you can solve this question like this. So these were questions based on rationalization. Now many people ask me question like this. I am taking one important question. So try to understand. Suppose x is equal to and many people have asked me this question now also. So I am taking this question x is equal to 3 plus root 2 divided by 3 minus root 2 and y is equal to 3 minus root 2 divided by 3 plus root 2. So you have to find out value of x square plus y square. So how can you solve this question? The basic way to solve this question is rationalize it. So there are people who do it without rationalization. There are people who do it with rationalization anyway you can solve it. I generally rationalize it. So this is 3 plus root 2 multiplied by 3 plus root 2 divided by 3 minus root 2 multiplied by 3 plus root 2 which is equal to 3 plus root 2 square which is 9 plus 2 plus 2 into 3 into root 2 which is 6 root 2 divided by 9 minus 2. So how much it is? It is 11 plus 6 root 2 divided by 7 and y would be equal to 3 minus root 2 divided by 3 plus root 2. So I will multiply the whole thing with 3 plus this is 3 plus root 2 divided by 3 plus root 2. Now how much it would be? This would be so this is just exactly opposite of this. So this would be nothing but 7 plus 11 divided by 6 root 2. Now how can you find out the answer of this x square plus y square? So you see here y is nothing but 1 by x. So what can be done here? So they are asking basically x square plus 1 by x square value which will be equal to x plus 1 by x whole square minus as simple as that. Now what is 1 by x? This is 1 by x. So one more method to do it without rationalization. One method with rationalization is you find out square of this. This is x so you find out square of this. This is y you find out square of this add both of them you will get the answer or what we can do is that let me go to let me make some space for you. Shraddha which one should be negative? I don't think anything should be negative here in this question. Okay one minute guys let me make some space for you. Second method to solve it is use this formula x square x plus 1 by x whole square minus 2. So what is x? x is 3 plus root 2 divided by 3 minus root 2. 1 by x is nothing but this this complete is 1 by x. So you find out 3 minus root 2 divided by 3 plus root 2 and you cross multiply it. So how much you get? You get 9 plus 2 plus 6 root 2 plus 9 plus 2 minus 6 root 2 divided by 3 minus 2 which is 3 square is 9 root 2 square is 2. So this is the value of x plus 1 by x. So how much this would be? This 6 root 2 6 root 2 gone. You get 9 plus 2 11 here and you get 9 plus 2 11 here. So how much do you get? 22 divided by 7. Now you find out square of this and from here you subtract 2 you will get your answer. So that is how you have to solve this kind of question. Okay now let me move to another question. Do you want similar kind of question to solve? Guys do you want similar kind of question to solve? Okay no issues. Okay I am solving another question which is very good. The same type of question x is equal to 7 plus 5 root 2. You have to find out value of x square plus 1 by x square is equal to what? So if x is 7 plus 5 root 2 1 by x would be how much? 1 by x would be 1 by 7 plus 5 root 2 which is nothing but 1 by 7 plus 5 root 2 multiplied by 7 minus 5 root 2 divided by 7 minus 5 root 2. So how much you get? 7 minus 5 root 2 in the numerator and 7 square is 49. 5 root 2 square is 25 into 250. So this is nothing but 49 minus 50. So this comes out to be 7 minus 5 root 2 divided by minus 1 which is equal to 5 root 2 minus 7. What what what is this? I have been doing this question for long time. So this is nothing but x plus 1 by x whole square minus 2. Now x is 7 plus 5 root 2. 1 by x is 5 root 2 minus 7. Square of this you take minus. So this and this one 5 root 2 plus 5 root 2 is nothing but 10 root 2. x square of this and you find and you do minus 2. So how much this would be? 10 square is 100 into 200. 200 minus 2 is how much? 198. So that should be your answer. So I have solved this kind of question. Now I am moving to polynomials and let me take off now in polynomials let me first discuss with you remainder theorem and what is remainder theorem? So remainder theorem tells that and this is very easy. You will definitely get one or two marks questions from here and if you know it you can solve it easily. Now try to understand. So remainder theorem tells that that if fx is a polynomial for degree greater than 1, so n greater than 1, n is what degree? So fx is divided by if fx is divided by x minus a then remainder would be equal to fa. What does that mean? If I ask this question Shreya is asking sir why do we subtract 2 from 10 root 2? What happens Shreya? Try to understand. I am solving a part of last question. You have been asked x square plus 1 by x square. So if you have to take this value what I am trying to do is that I am taking x plus 1 by x whole square which is x square plus 1 by x square plus 2 into x into 1 by x. So this xx gets cancelled out. Now if you have to find out only this value which is x square plus 1 by x square that would be equal to take 2 on this side. So this would be x plus 1 by x whole square minus 2 is equal to x square plus 1 by x square. So I don't want to solve this value because this is complicated. I want to solve this value because x plus 1 by x gets cancelled out. Few things get cancelled out. Like in last question it was 7 plus 5 root 2 plus 5 root 2 minus 7. So 7 7 got cancelled out. I was only left out with 5 root 2 and 5 root 2. That's why I am trying to solve it in this fashion rather than getting a square of it and adding it. So I hope you understood what I am trying to say. Now let me again move to remainder theorem. If I talk mathematically remainder theorem so in form of expression suppose I have a quadratic expression so which is if I am asking you that suppose fx is equal to x to the power 4 minus 3x square plus 2x minus 3 is divided by x minus 2. What would be remainder? You have to find out remainder. So what do you do here? You put x minus 2 is equal to 0. So x comes out to be 2 and put this value x is equal to 2 here. So 2 is how much? So f2 would be equal to 2. Actually you have to find out f2. f2 would be remainder. See here if it is divided by x minus a, fa is remainder. So divided by x minus 2 what I am doing? I am putting x minus 2 is equal to 0. Whatever is the value coming? I am putting in the expression. So 16 minus 3 into 4 is 12 plus 2 into 2 is 4. This is nothing minus 3. So how much it is coming? 16 minus 12 is 4. 4 plus 4 is 8. 8 minus 3 is 5. So what would be remainder? Remender is 5. So such questions can come. Now few other things of remainder theorem. I will just quickly wrap up through it in next 2 to 3 minutes. So I will try to revise most of the things which are part of syllabus by taking good questions. So now try to understand what is a factor? Factor is something which divides it completely, which divides any expression completely. So if I am saying that x minus a is factor of a x to the power n plus v x to the power n minus 1 plus c x to the power n minus 2 and likewise. So what do you mean by factor? Factor means complete division. Complete division means what? Remender should be 0. It means that what is the remainder? If you divide f x by x minus a, remainder is f a and if it is a factor and remainder is 0, it means that if x minus a is factor then f a would be 0. So if f a is 0, so if I say you that prove that x minus 3 is factor of x cube plus x square minus 17x plus 15. What do you do? You put x minus 3 is equal to 0. Value of x comes out to be 3. You put it in this equation. So this comes out to be 27 plus 9. 17 into 3 is 51 plus 15. So 27 plus 9 is 36. 36 plus 15 is 51. 51 minus 51 is how much? 0. So it means that f of 3 is coming out to be 0 when this is getting divided by x minus 3. It means that x minus 3 is a factor of it. Now what kind of question is asked here? The question here is like this. The question asked here is if x minus a is factor of x to the power 5 minus a square x cube plus 2a plus 2x sorry plus 2x plus 2x plus a minus 3 then find a then find a. Now what happens? Put x minus a is equal to 0. So x comes out to be a. Put this value here. So this is what? a to the power 5. This is a square multiplied by a cube plus 2 into a plus a minus 3 and if it is factor then this remainder would be what? 0. So see here a square into a cube would be how much? a to the power 5. This and this will get cancelled out. So 2a plus a is how much? 3a. 3a minus 3 is equal to how much? 0. So a comes out to be how much? 3 by 3 which is equal to 1. So that's how you have to solve these kind of questions. Now I'm moving ahead from remainder theorem. I'm going towards some other topic. So again I'm taking I'm going to factorization now in polynomials and I'll solve two three questions from factorization and then I'll move ahead. Two questions. So look at this. This is the factorization. I'm asking question factorize what do you want need to factorize? 2 root 2 a cube plus 8 b cube minus 27 c cube plus 18 root 2 a b c. Now by looking at it we can say that this is nothing but a cube plus or let me because a b c has been given here. Let me write it in some other term. So this can be written as p cube plus q cube plus r cube minus 3 p cube r and it can be factorized. This is this format and it can be factorized at factorized as p plus q plus r multiplied by p square plus q square plus r square minus p cube minus q r minus r p. So what is the value of p? Let's compare it. p cube is equal to 2 root 2 a cube. So p would be root 2 a cube cube is equal to 8 b cube. So q would be equal to 2 b and r cube is equal to minus 27 c cube. So r would be equal to minus 3 c. So let's put the value here. It will be root 2 a plus 2 b minus 3 c and p square would be how much? So root 2 a square would be 2 a square. q square would be 4 b square. r square would be plus 9 c square. I am writing here p into q is 2 root 2 a b and with a negative sign. So minus 2 root 2 a b. Now q r is already negative here and r is negative. So negative negative will become what? Positive. So I am writing here positive 2 b into minus 3 is actually minus 6 b c one negative sign here. So this is 6 b c and here also 2 root a into minus 3 c is minus 3 root 2 a c and plus a negative sign. So this becomes positive. So 3 root 2 a c. This is what is rationalization. So I hope you all understood it. Let me move to some other type of question. Now one type of question which is primarily asked is like this that p is one kind of question is this p is equal to 2 minus a. Then prove that a cube plus 6 a p plus p cube minus 8 is equal to 0. So what I need to do is that I need to take it on this side. So a plus p minus 2 would be equal to 0 and this is what this looks something like p cube plus q cube plus r cube minus 3 p cube r. Now try to understand when you open the bracket or you find out the factor of it. The first factor is p plus q plus r and if p plus q plus r is 0 then this becomes 0. So if p plus q plus r is 0 then p cube plus q cube plus r cube minus 3 p cube r is equal to 0. Now see here a plus b plus c is 0. So a cube plus b cube plus c cube minus 3 a b c would be 0. So what is a cube a cube b cube is p cube c cube is minus 8 minus 3 a b c one negative is there. So that becomes positive. So 3 into 2 6 a p that is equal to 0. So just write this formula and show this you will get your answer. Now let me solve some other type of question. Okay so let me write a question for you and I will give you one minute of time to solve it. I will not immediately solve it. So I am not immediately solving it. I will give you one minute of time to solve it and that is look at here x by y plus y by x is equal to minus 1. It has been given that x not equal to 0 and y is also not equal to 0. Then find the value of x cube minus y cube. So take this question and solve it for one minute and give me the answer. What is the answer? As soon as I gave a few questions few of you just left the session and after this I am moving to Heron's formula. Now I hope you all are sitting with your RS Agarwal book because most all of you have RS Agarwal book. So now I am solving it cross multiply it so you get x square plus y square divided by x y and that is equal to minus 1. So cross multiply you get x square plus y square is equal to minus x y which is x square plus y square plus x y is equal to 0. Now when you open the bracket this can be written as x minus y and x square plus y square plus x y. Now this is equal to what? 0 from where I have proved it here. So x minus y into 0 is equal to 0. So both Gourman and Shreya you got the right answer. The answer is 0. Now let me just check whether I can give you some other question also. Okay a good question just let me raise it. For this question also I will give you one minute of time. Let me check how many of you can solve it. So question is okay Anurag is asking and Nihar is asking how did I solve? So x by y plus y by x was given to be minus 1. So cross multiply it. So LCM is anyway x y so this would be x into x x square this would be y into y y square. So x square plus y square divided by x y is equal to minus 1. So what I do? x square plus y square and multiply this x y to this minus 1. This will be x square plus y square is minus x y. So take it on the other side you can write it x square plus y square this goes on the other side becomes plus x y and that is equal to 0. Now you have to find out x cube minus y cube the value is what? So it can be written as x minus y and x square plus y square plus x y. Now what is this value? I have already proved for this scenario x square plus y square plus x y is 0. So I am putting this value equal to 0. So x minus y multiplied by 0 will give you how much? 0. So what is the value of x cube minus y cube? x cube minus y cube is coming out to be 0. So Anurag and Nihar did you understand now? Did you guys understand Anurag and Nihar? Anurag is saying yes Nihar what about you? Okay great so I am moving ahead. I am giving you one or two more questions. So please write it down. Okay look at here a plus b plus c is equal to 0 then what would be the value of a square divided by bc plus b square divided by ca plus c square divided by ab. You have to find out the value of this. How many of you can solve it? Just take one minute and solve it and one more question and I am done with polynomials. I will move to Heron's formula. Okay Nishant is saying the answer is 3, Eshani is saying the answer is 3, Shreya is saying is it abc, NPS Rajaji Nagar, are you guys there or not? Anurag is saying 3, Gourman is saying 3. Shreya just check your answer your answer is not right, Shraddha is saying 3. Okay great so most of you are getting the correct answer. Now see here if a plus b plus c is equal to 0 then I can write I just discussed it a cube plus b cube plus c cube minus 3 abc is also 0. So what I can write a cube plus b cube plus c cube is equal to 3 abc take this abc on this side so you will get a cube plus b cube plus c cube divided by abc that would be equal to 3. So now let's see here the LCM of this would be abc now this is bc so a would be left out so a square into a is a cube this is ca so b would be left out b square multiplied by b would be b cube ab so c is left out this is c cube so if abc is equal to 0 a cube plus b cube plus c cube divided by abc I have just to prove here how much it would be it would be 3 so that's your answer now last question of polynomial I'm putting okay now this question also solve it x plus y plus z is equal to 9 xy plus yz plus zx is equal to 23 so find out the value of x cube plus y cube plus z cube minus 3 xyz what would be value of this so take two minutes and solve this question and give and and shoot the answer okay I got a few answers which is 108 now look at here how to solve it so just find out the fraction factor so x plus y plus z and x square plus y square plus z square minus xy minus yz minus zx now I need to shot this out because the value of this has not been given so x square plus y square plus z square see how I find the value so I find out this value x square x sorry this is x plus y plus z whole square is equal to x square plus y square plus z square plus 2 xy plus 2 yz plus 2 zx now if you have to find out only this value this 2 xy 2 yz 2 zx will have to go on the other side so x square plus y square plus z square I am writing x plus y plus z whole square minus 2 xy minus 2 yz minus 2 zx so look at here already this particular value is minus 2 xy minus 2 yz and minus 2 zx and you have extra minus xy minus yz minus zx so I can simplify this to x plus y plus z and this I can write as this complete I can write as x plus y plus z whole square this minus 2 xy minus 2 yz and minus 2 zx in combination with minus xy minus yz and minus zx would be nothing but minus 3 xy minus 3 yz and minus 3 zx now let me put value here x plus y plus z is how much so this is 9 now this is how much so 9 into 9 is 81 square minus 3 xy plus yz plus zx is 23 so I have taken 3 common from here I will make it 23 so this is nothing but 9 81 minus 69 23 into 3 is 69 which is equal to 12 so 9 into 12 is how much 12 into 9 is equal to 108 so answer is 108 most of you have got the answer now let me move to a few herons formula question so give me a moment I just teach you heroes formula where is it so I am taking a difficult question like this so first of all let me write down what is heroes formula heroes formula is nothing but is any triangle has been given a b and c so area of this triangle would be under root s s minus a s minus b and s minus c where s is semi perimeter of the triangle so perimeter is summation of length of side abc that is divided by 2 so this is the herons formula now how do I use this formula so always remember area of triangle is given by so many like if right angle triangle is there and this is perpendicular and this is base so area would be this is 90 degree area would be equal to 1 by 2 into perpendicular into base if I don't give base I'll have to give hypotenuse where relationship is hypotenuse square is perpendicular square plus basis square now try to understand if I give an equilateral triangle this is a this is a this is a area would be equal to root 3 by 4 a square if I give isocellous triangle suppose this is a this is a this is b so how do you find out the area guys I'm proving it try to understand the generalistic formula of area is any triangle has been given and this length of the altitude has been given suppose I hide which is called h is given and you know the base area of triangle would be 1 by 2 into height into base this is the most generalistic formula of any triangle now try to understand if I draw a perpendicular here from this this particular vertex would be non equal vertex because these are the two equal sides so angles opposite to it would be equal angle so I call it equal vertexes and this is non equal vertex so if from non equal vertex you are drawing a perpendicular on the non equal side it would be perpendicular bisector also so this is what this is b by 2 now this hypotenuse is given so this height would be equal to a square minus d by 2 whole square so how much it is this is nothing but a square minus b square by 4 so I write it as 4 a square minus b square divided by 4 so which is equal to 1 by 2 under root 4 a square minus b square so height is equal to 1 by 2 under root 4 a square minus b square so if I have to find out area area would be 1 by 2 into base base is how much b and how height is how how much 1 by 2 into 4 a square minus b square which can be written as b by 4 under root 4 a square minus b square this has not been given at most of the places in your book so please pay attention to it so area of isocellar triangle I write like this on equal side divided by 4 minus 4 time equal side square minus on equal side square as simple as that on equal side divided by 4 under root 4 into equal side square minus on equal side square so remember these formulas these are 4 5 formulas which would be used invariably in heroes formulas theory have you all written it I'm giving you 30 seconds of time so that you can write it and if you have any doubt just post it so that I can explain you and then I'll move ahead okay now let me solve a few questions so question is isocellar triangle base is equal to 24 centimeter so you have to find out an area has been given as 192 centimeter square you have to find out perimeter so what do you mean by perimeter you already have non equal side with you which is 24 centimeter if you if you are able to find out equal side you will be able to find out perimeter so suppose this is a this is a and this base is how much 24 and this is how much h now 1 by 2 into h into 24 is equal to how much 192 so h is equal to 192 into 2 divided by 24 so this is 8 8 into 2 is 16 so h comes out to be how much 16 so this is 16 now try to understand I told that this h will divide wherever the if I'm drawing h from this non equal vertex it will divide the base into two equal parts it will be perpendicular bisector so this would be 12 so h square plus 12 square would be equal to a square so a is coming out to be how much a would be equal to h square h is 16 so 16 square is 256 12 square is 144 and that is equal to how much a square so a would be how much 256 plus 144 is equal to a square is equal to 400 and a comes out to be how much 20 so this is how much 20 this is how much 20 this is how much 24 perimeter would be how much 20 plus 20 plus 24 how much it will be 64 so that is your answer now let me move to another question now instead of in in heron's formula generally what they do is they give it in quadrilateral format and which quadrilateral format just I'll explain you that quadrilateral and and perhaps then you will able to solve questions best on that so look at here suppose this is one quadrilateral something like this and I draw a diagonal like this suppose this is my diagonal so suppose this is a this is b this is c this is d so what happens I draw a perpendicular from here and I draw a perpendicular from here so suppose this is b x and this is d y so area of quadrilateral would be quadrilateral would be area of triangle summation of area of triangle abc plus summation of area of triangle I am writing area here area of triangle acd as and what is it so area of a abc would be 1 by 2 base is what ac multiplied by what bx plus 1 by 2 into area of acd would be 1 by 2 into ac because base is same multiplied by what dy so what is it if I say bx is equal to h1 and dy is equal to h2 2 heights so I can write this h1 and I can write this h2 so what I can do I can take out 1 by 2 ac outside and in the bracket what would we left out h1 plus h2 so many a times this formula is used in heroes formula if you know this heroes for this formula you can solve difficult question best on heroes formula now let me take one question where something like this has been made so something like this has been made I am making it just for one of them this is a tile which has been divided into two equal let me make a bigger one so that this is a tile and what happens is you have been given this is 20 this is 16 and this is 12 centimeter so you have to polish it you have to do some painting on it cost of painting is one centimeter rupee sorry rupee one per one centimeter square or per one per centimeter square so what do I do triangle area can be found out how can you how can you find out triangle area so s is equal to 20 plus 16 plus 12 divided by 2 which is equal to I'll solve it with another another method also which is equal to 32 sorry 20 plus 16 36 36 plus 12 48 divided by 2 which is equal to how much 24 so you put it in herons formula so 24 and s minus a so 24 minus 20 is how much 24 minus 16 is how much 24 minus 12 is how much 12 so how much this would be this would be 24 into 4 so let me write it how much this is this is 4 is 3 into 8 this is 4 this is 8 and this is 3 into 4 so 2 4's I take out 1 4 2 8's I take out 1 8 and 2 3's I take out 1 3 so how much it is 3 into 8 24 24 into 4 how much it would be 96 so this is nothing but 96 centimeter square so just to cover this area this portion one triangle is nothing but 90 this is 96 centimeter square and for one centimeter square I'm charging one rupee so I would be spending how much amount of money I'll be spending 96 rupees now was there any need to utilize heroes formula here I don't think so why because we just utilize this guys understand this triplet 2016 12 so 2016 12 this triplet is a triplet of right angle triangle how you see here this is 3 this is 4 this is 5 is 345 if I give you 345 perhaps you'll understand that triplet as right angle triangle and you will find out that this is 4 this is 4 this is 3 or vice versa and this would be 5 so area of triangle would be 1 by 2 into 3 into 4 how much that would be that would be 6 now what I have done in this triplet try to understand this is one of the most basic triplet of right angle triangle 345 you multiply it with 2 you get 6 8 10 multiplied with 3 you get 9 12 15 multiplied with 4 you get 12 16 20 so wherever such kind of triplet is coming you should utilize the formula of right angle triangle directly here heroes formula was to be used I showed you by using heroes formula otherwise you can directly write 1 by 2 into 12 into 16 and how much this would be 16 into 6 is 96 so 96 centimeter square multiplied with 1 you'll get 96 rupees so that's how you could have solved this question now let me solve one or two more questions let me solve one more question and then I'll wrap up the class because we have done one more question from heroes formula and then I'll wrap up the class so if if you have if you are given 10 12 or 15 such kind of figure don't get worried about that solve it as you can so let me draw a diagram for you the diagram is like this so that diagram is like this so they have given that this is 17 this is 15 this is 90 degree this is 9 and this is 12 so what what should be area of this coordinator so what I'll do is that I'll find out here in this space what this base would be under root hypotenuse 17 square is 289 minus perpendicular which is 15 square is how much 225 so which is equal to under 64 that is nothing but 8 so how much this would be this would be 8 so in this particular part I don't need to utilize herons formula 1 by 2 into 15 into 8 so this would be how much 15 into 4 that is equal to 60 and here I can utilize this is also nothing but 345 thing so you see here 345 multiply it with 3 so you get 9 12 and 15 if question is adamant that you have to apply herons formula over here what you can do is you can so let me solve this with herons formula you can also do because I have already proved that this angle would be right angle triangle you can do it with 1 into 1 by 2 into 9 into 12 which is 9 into 654 so you add 60 plus 56 which is 116 if you want to solve it with herons formula s would be 9 plus 12 plus 15 divided by 2 which is equal to 36 divided by 2 which is 18 so put it in the bracket 18 18 minus 9 is 9 18 minus 12 is 6 18 minus 5 is 3 so how much that is let me write it this is nothing but 3 into 6 this is nothing but 3 into 3 this is nothing but this is 2 this is 6 and this is 3 so 4 3 and 1 6 so I take out 2 3 and 1 6 so 9 into 6 is 54 so again 60 plus 5 how much I have written guys 54 so this is not 140 116 this is a 114 so 60 plus 54 how much it would be 140 anyhow you can solve this kind of question okay so guys I am wrapping up the session now if you have any doubt you can send send me message on my personal chat or whatsapp group so thank you so much for attending the class