 Okay, good evening, everyone. Let me start now. Tehran's formula, as we already know, is the area of triangle. So I will just write area of triangle. I would not get into theory. This session is mostly about problem solving. And what we will do here is we will solve a lot of questions. So if you have any doubt, you can share it with me at any point of time. And then we'll solve questions out here. Let me write the first helical. Triangle. We have triangle something like this. This I mark as C. Side of length A, side of length B and this is the side of length B. The semi-perimeter would be plus B plus C divided by 2. And triangle under root S. S minus 1, S minus C. This formula is nothing but... Call this formula as Tehran's formula. Starting with this question. The question is something like this. Triangular park. Write down the question. A triangular park, A, B, C, sides 120 meter, 80 meters and 50 meters. A gardener has to put a fence all around it and also plant the grass inside. Find the area planted. Find the cost of fencing. A rupees 20 per meter. Leaving a space for 3 meters wide gate on one side. Started getting few answers. Okay, few answers I have been getting. Let me... Okay, I'll give you one or two more minutes. One answer everyone is giving right. Fencing answer, most of you are giving right. The area answer, none of you, I didn't get only from... Let me check. One person, I have got it right. The area of answer given by you, except one person is not right. Okay, let me solve this question. First, let me find out the area. So, area would be A is equal to 120, B is equal to 80 and C is equal to 50. So, S is equal to 120 plus 80 plus 50 divided by 2. Okay, now just focus on the answer part rather than giving the answer. So, S is 125, so I am writing it as 125, then it is 125 minus 120. 125 minus 80 and 125 minus 50. So, this is 125 into 5 multiplied by this is 45 and this is what? 75. Now, generally there should be a case of prime factorization. So, this is 5 into 5 into 5, then I have 5. So, I have 45 is 3 into 5 into 5 and 75 is equal to 3 into... Sorry, this is 3 into 3 into 5 and this is 3 into 5 into 5. So, how many 5s here? 1, 2, 3, 4, 5, 6, 7, 8. No, sorry, 7. So, 3, 4, 5, 6, 7. So, out of 7 5s, 3 5s will come outside. So, 3 5s are here. Then how many 3s? So, I have 1, 2, 3, 3. So, out of 3 3s, 1 3 will come outside. So, inside the square root, what will remain? 1 3 and 1 5. So, I will have 125 into 3 3 75 and in bracket, I will have root 15 and don't forget to write meter square. Now, let me go and give you a question. Sorry, let me solve the second part and then give you a question. So, second part is fencing is done on the perimeter. Now, what is the perimeter? s is equal to 120 plus 80 plus 50 that is equal to 250. Out of 250 meters, 3 meters are left out for the gate. So, total fencing is done on 250 minus 3. That gives me 247 meters and 20 rupees is needed. So, cost of fencing would be equal to 247 into 20 which gives me 4940 rupees. So, that's the answer. Now, let's move to second question. Sorry, second question which is just give me a moment. So, second question would be, so suppose I am taking a question where you will have to find out altitude from the area. So, it's something like this. So, let me make a triangle or let me give you a different question altogether. And in this only, suppose I draw a line something like this and then it is something like this. So, my question is that this is 120 meters. This is 122 meters. This is 22 meters. This is 24 meters and this is 26 meters. You have to find out area of this shaded region. Just solve it. Find out the area of the shaded region. Did you solve it? I'm not getting any answers till now. Let me know in the chat box what's your answer. I give you 2 to 3 minutes more. I intend to solve at least 10 questions in the class. 10-12 questions. Sarduli, the question is written on the board. So, you have to find out area of the shaded region. Okay, no issue, Rishit. I will do it. Don't worry. Try to find out the values of... A lot of people are giving me answer root 35, root 105. Why can't you find out the values of root 105 and root 35? You can definitely find out value of root 105. No, 673.2 is not the answer. No, 60 root 138, I don't think it's the right answer. 13505 is not the right answer. Okay, let me solve this question. Yeah, 10, Rishit 1074 is the right answer. Good. So, we have S is equal to... So, let me name this. This is A, this is B, this is C, and this is P. So, area of triangle, ABC I'm finding, S is equal to 120 plus 122 plus 22 divided by 2. So, this gives me 120 plus 122 is 242 plus 22, 264 divided by... So, this is 264 divided by 2 is 132. Now, I will have to find out area. So, it would be formula S, which is 132. S minus A is 132 minus 120, then 132 minus 122, and then I have 132 minus 22, everything inside the square root. So, this gives me nothing but 132 into 12 into... How much is this? This is 10, and this is how much? 132 minus 22 is 110. So, I will not do the prime factorization. I know that this is 12 into 11. This is 12 already here. This is 10, and this is 10 into 11. So, I have 12, 11... Sorry, two 12s, two 11s, and two 10s. So, one 10, one 11, and one 12 come outside. It gives me an answer of 1320 meters square. Now, area of second triangle would be equal to... That's the wrong answer, 136080. Area of second triangle is... So, area of triangle PBC, I will have to find out. And for that, S would be equal to 24 plus 26 plus 22 divided by 2. So, how much it gives me? It gives me 72 divided by 2, and that's nothing but 36. So, I utilize this value of S here in my formula, 36, 36 minus 24. Then I have 36 minus 26. And then I have 36 minus 22, which is equivalent to 36 into 12 into 10 into 22. And that is nothing but... If you look at it, it will give you... 36 will come out at 6. Now, 12 can be written as 2 into 2 into 3. Then I have 2 into 5. And then I have 2 into 11. So, I have how many 2s? I have 2, 4 2s. So, 2 2s come outside. So, 6 into 4. I mean, 4 comes out. So, it becomes 24. Then I have 3 into 5 into... What do you say? So, 1 seconds, 36, 24, 26, and 22. So, 24 divided by 12 and 22. No, guys, this is wrong. 36 minus 22 will give me 14 here. So, I am writing it 14 here. And this gives me nothing but 14. So, just give me a moment. So, this gives me 2 into 7. So, no change here. Just that 2 into 7. 3 into 5 into 7 will give me 105. Now, I need to find out what is the square root of 105. And if you will see square root of 105, if you try to find out, as you already know, square root of 105 would be equal to nearly 10.25. So, 24 into 10.25 is nothing but... Yeah, I have corrected. It's nothing but 24 into 5 is 120. Then 24 into 2 is 48 plus 12, 60. So, 6. And then I write 6 here. And then what do I get? So, I get 246. So, I have 13320 minus 246. So, this gives me nothing but... I have 4 here. And then I have 7 here. And then I have 0 here. And then I have 1 here. So, it's 1074 meter square and nothing else. So, I hope you understood this question. Now, let me give you another question. The other question is, let me make a parallelogram for you. So, let me make a rectangle for you. Okay. This is a rectangle and which I'm drawing this line. And now listen to the question carefully. So, I am saying that this is a rectangle ABCD. So, we have a rectangle ABCD. ABCD is a rectangle where this length is 51 and this length is 25. So, this is nothing but dimensions are 51 into 25 centimeters square. Now, I draw a line PQ such that PQCD is a trapezium. So, I have a PQCD trapezium and in which QC is parallel to PD. And QC by PD is equal to 9 by 8. If area of PQCD divided by area of ABCD is equal to 5 by 6, then find out PD and QC. Now, here I don't think you will have to use Heron's formula. Solve it quickly. Two, three minutes, one more than that. I've started getting few answers. There has to be two values, length of QD and PC. I'm not asking summation of them. Okay. Few answers I'm getting. Okay. Try to understand. Let's find out this first. Okay. So, I'm going to solve it now. Let's find out what is the area. So, if you look at it, area of PQCD would be equal to, area of trapezium is equal to 1 by 2, sum of lengths of parallel sides multiplied by height. So, here it will be 1 by 2, QC plus PD. And if you look at the height, height is this DC. DC divided by area of triangle ABCD would be equal to DC into DC. And that is equal to how much? 5 by 6. So, I can write this as 1 by 2 QC plus PD divided by, because DC gets cancelled out, divided by BC is equal to 5 by 6. So, QC plus PD divided by BC is equal to 5 by 6 into 2, which is equal to 5 by 3. So, from here QC plus PD comes out to be 5 by 3 into BC. And BC has been given as 51 meters. So, 5 by 3 into 51 gives me nothing but 17 into 5. QC plus PD is equal to 85. I have been given QC divided by PD as 9 by 8. So, if I assume QC to be 9K and PD to be 8K, I keep these values here. 9K plus 8K is equal to 85. So, 17K is equal to 85 or value of K is equal to 5. If value of K is 5, then it gives me 9 into 5, 45 and 8 into 5, 40. So, answer here is 45 and 40. Now, let me give you another question, which is, suppose I have a parallelogram, something like this, and this is the diagonal of the parallelogram. The parallelogram is named as A, B, C, D, where DC is, or suppose this is 12 and this is 17 and this is 25. So, suppose this perpendicular is drawn and suppose this is called M. So, you have to find out area of parallelogram also length of altitude, length of altitude, AM. Okay, am I getting few answers? Let me solve this question for you. So, first of all, we know that the diagonal of the parallelogram divides the parallelogram in two equal halves. So, if I find out area of triangle, area of triangle, DBC. So, area of parallelogram is equal to two times area of triangle, DBC. So, area of triangle, DBC is nothing but you find out S for it. That is 25 plus 12 plus 17 divided by 2. 25 plus 12 is equal to 37 and 37 plus 17 is nothing but 54. So, 54 divided by 2 is equivalent to 27. So, area would be under root 27 into 27 minus 25 into 27 minus 12 into 27 minus 12 into 27 minus 17. So, this gives me 27 into 2 into 15 into 10. So, how much it comes? Let me prime factorize it and this gives me 3 into 3 into 3 into 2 into 3 into 5 into 2 into 5. So, I have 4 3s. So, 2 3s comes outside. I have 2 2s. So, 1 2 will come out and 1 5. So, 1 5 will come out. So, this gives me 9 into 2 into 5. Sorry, 2 into 5. So, 9 into 2 into 5 is nothing but 90. So, this is 90. So, area of parallelogram would be equal to 2 into 90 which gives me nothing but 180. Now, area of triangle is also 1 by 2 into height AM into base based on which the altitude has been drawn. So, this is equal to BC here and that is equal to 180. So, 1 by 2 into AM into 12 is equal to, sorry, here I am finding out area of parallelogram actually. So, if you have to find out area of parallelogram, so you don't need to write 1 by 2 here, it will be base into height. So, what you need to do is area of parallelogram is nothing but AM into BC and AM, that gives me AM is equal to 180 divided by 12. So, the answer is 15, what is the value? 15 meters, right? So, answer is 15 meters. Okay. Another question is, write down, what is this chat all about? Okay, people are giving me answers. Write down a rhombus, I'll leave it. This is not basically Heron's formula. So, okay, let's do a question of proving. This is not difficult but you'll have to prove it. If each side of a triangle is doubled, then find the ratio of area of new triangle thus formed. You need to solve it. I also know the answer. So, don't try to become too intelligent. Apply it, do it with the help of Heron's formula. Those who are giving me answers, I know the answer. Don't give me answer. Solve it. Use Heron's formula. Okay, I know that you all will give this answer quickly. So, let me give you another question. I'm not solving this question. Another question is a rhombus sheet with perimeter 40 centimeter and one diagonal 12 centimeter is painted on both the sides at the rate of rupees 5 per meter square. Find the cost of painting. Okay, so rhombus is something like this. A lot of you are giving me answers now. So, if perimeter is 40, so side would be 10 because all sides are same. So, AB is 10. Now the diagonals are the perpendicular bisectors of each other. So, if one diagonal is 6, 12, so half of the diagonal would be if BD is equal to 12, BO would be equal to 6. So, AO square plus BO square is equal to AB square because diagonals are perpendicular bisectors. So, AO square plus 6 square is equal to 10 square. So, AO square is equal to 10 square minus 6 square which is equal to 8 square. So, AO is equal to 8. So, now area of rhombus is 1 by 2 into D1 into D2. So, that is 1 by 2 into 12 into, so this AC would be equal to here 2 into 8 which is 16. So, 12 into 16. So, this is nothing but 6 into 1696 and you multiply this with 5. So, that gives you 480 rupees. Next question is the perimeter of a triangle. Write down the question. Don't chat. The perimeter of a triangle, the perimeter of the triangle is 50 centimeters, is 50 centimeter. One side of triangle is 4 centimeters longer than the smaller side. And third side is 6 centimeters less than twice the smaller side. Find the area of the triangle. I don't like this way of giving the answers 20 root 30. Why can't you find out the value of root 30? All are CBSE students. Where you will have to write these answers without any root. Okay. So, let me solve this question for you. Now, the previous question, the answer is 480 only. It's a 2D figure. So, whether it is two-sided or one-sided, the answer remains 480. Rhombus is not something like a cuboid where if you paint two sides, it's a different painting. It's a 2D figure. Whether you paint it on one side or two sides, it remains the same. Now, if the smaller side is X here, then the second side is 4 centimeters larger. So, X plus 4. And third side would be 6 centimeters less than twice. So, 2X minus 6. Now, perimeter is 50. So, X plus 4 plus 2X minus 6 is equal to 50. So, this gives me 4X minus 2 is equal to 50. So, 4X is equal to 52. So, X comes out to be 13 here. So, second side would be X plus 4 would be equal to 17. And the third side 2X minus 6 would be equal to 2 into 13 minus 6, which is equal to 20. So, S is equal to 20 plus 17 plus 13, which I already know the perimeter is 50. So, that gives me 25. So, area of triangle would be equal to S, S minus A, S minus B, S minus C. So, that is equal to 25. Then I have 25 minus 13. Then I have 25 minus 12. Sorry, how much it is? 25 minus 17. And then I have 25 minus, how much I have written here? 25 minus 20. So, this gives me 25 into 12 into 8 into 5. So, this gives me nothing but 5 into 5 to 2 into 2 into 3 into 2 into 2 into 5. How many 5s? 1, 2, 3. So, 1, 5 comes outside. How many 2s? 1, 2, 3, 4, 5. So, 2 2s come in outside. And then I have 1, 5, 1, 2 and 1, 3. So, I have nothing but for 20 root 5 into 10 into 3 root 30. So, I have to find out value of root 30. Root 30 is nothing but 5.48. So, this will give me 54.8 into 2, which is nothing but 109.6 cm2. So, that's your answer. So, this is the answer. Somebody is asking me how to find out the value of root 30. You write like this. So, this is root 30. Now, how many times it goes? It goes 5 times. So, 5 into 5 is 25. So, 5 comes here. 5 plus 5 is 10. Now, there is nothing outside. So, I put a 0 and then 2 0s will come here. So, this will go how many times? 4 times. So, 104 into 4 is 416. So, this gives me 84. And I add this 4 here. This 4 here. So, this gives me 108. Now, here comes 2 0s. So, it goes 7 times. So, 7 into 7 is 49. And then, I have 7 into 8 is 56 plus 4 is 60. So, 6. And then, 6 comes here and then 7. So, I have a likewise a coupon doing. So, you have 5.47. Again, you will do. You will get another 7 perhaps here. And that converts into 5.48. So, this is it guys. This is it in today's Centrum R class of Mathematics. Thank you so much for joining me for this class. And I hope you enjoyed the class. We will keep on solving questions in Centrum R classes regularly. So, thank you so much for joining me and see you next time. Thank you.