All right, guys, it's time. Welcome to another MetConversation. So today we're going to talk about 2D geometry. And please work on the question. Please share your answer via chat. And I see some of our friends are joining. Thanks so much. Let's see. So far, one of our friends says C. I'm not going to say if it's right or wrong, but let's see what about other friends. Just start the timer for you. How many points do the diagonals of a regular hexagon intersect? We do not count vertices. We have hexagonal shape, guys. I recommend you to work with some sort of hexagonal shape. It's not going to be perfect, but you've got the idea. It has been 1 minute and 20 seconds. We get only one answer, which is C. What about others? I should get a couple of more answers, guys. I believe you remember what diagonal means. Those lines we call diagonal. One friend said answer should be d. Okay. Thanks, Som. I will put that already. Thanks, buddy. All right, guys. At the end, you are going to count those points. I don't know why do I get only two answers, by the way. This part should be, you know, only one point. The shape is not perfect, so it's kind of seem different. After you draw all of those Little diagonals you gotta count to find how many points do they intersect Let's start guys. It has been almost five minutes. You just count one Two three Four five six Seven eight nine Ten Eleven twelve and thirteen guys Whoever said 13 that friend. Yes, right. Thank you so much All right guys answer was 13 as you see is there any question about this one That means no we can move on all right A little recap about today we gotta talk about to the geometry guys so far We had calculations number theory algebra working. I do algebraic thinking work rate ratio and statistics A Problem when you solve the problem just a little problem little recap make sure you understand it You need to plan you need to carefully complete your calculations and Reflect it's supposed to be some sort of answer which makes sense for the situation To Dimensional geometry involves all problems with shapes or objects that occur in a flat plane for example Problems can include lines segments and rays These can be parallel or intersecting Met Kangaroo may ask you to use your knowledge about angles formed by these Objects a right angle measures 90 degrees Straight lines measure 180 degrees. These are examples here as you see and the complete circle measures 360 The sum of angles in a triangle is 180 You can use the equal angles and sides of a Equilateral and isosceles triangle to help solve many problems. We're gonna mention about that. No worries Right triangles have only one 90 degree angle the lengths of the sides have a special relationship we call them a square plus b square equals c square where C is the hypotenuse and Other legs other sides are a and B. We call them legs the right sides. This is known as Pythagorean theorem After I just explained the slide guys if you have any question, just ask me please if you need some clarification I think In the next one you should be familiar with the variety of different 2d shapes Close polygons have straight sides and are named by number of their sides As you see we have pentagon five sides hexagon six sides and so far In the regular polygons the sides and the angles that are all the same size at each other If you are talking about regular ones, by the way Many polygons have polygons have special names based on the parallel sides Concurrent sides right angles or other distinctive features. You should have studied these previously a Little German a little vocabulary about geometry congruence means two or more objects have the same dimensions, basically Similar two or more objects that have the same shape and the angle measurements, but they differ by scale factor That means one of the object is basically a little bigger. Another one is smaller Bisector the bisector of a line segment cuts it into two concurrent pieces Bisectors that intersect the original segment at a 90 degree are perpendicular bisector The bisector of an angle is a ray that cuts the angle into two congruent house When we see bisector basal guys that line divides either angle or side by two Same measurements Whenever I take the cuts an angle in a triangle It divides opposite sides into the ratio of the measure of the other two sides basically Here for a given triangle ABC guys, we can say DB here Over DA We can just stop that Proportion from angles as you because we have same angles since CD is bisector That measurement is supposed to be called CB here Divided by CA The three angle bisectors intersect at the in center of a triangle In center basically when we work with the circle when we just put a circle which fits inside of triangle the center of You know, that's circle also called in center We can move on there is no question so far Parameters and areas the perimeter of a 2d shape is a distance around all sides The total length of all the segments and curves which make it is other order The perimeter of a circle is called also circumference and it's calculated by the formula 2 pi R or 2 D Basically 2 or pi D basically pi times diameter where R is the radius of the circle Area is the amount of two dimensional space a figure covers As you see when we talk about perimeter guys, we are talking about something around of the shape But when we talk about area in that case we are looking at how to fill in that two-dimensional shape As you see Area of the rectangle basically the basic formula length times width Since triangle is half of the rectangular shapes we can say half times space times height or half times length times width basically an Area of circle we gotta use the formula of pi R squared All right, so far no question. We can just move on jumping and discussing some questions and this is my first gift for you from 2016 and number 5 Jonah turns a car over about its lower edge and then about its right edge. It's shown What does she see at the end? So this is the car as you see guys She just turned over here Just follow the arrows as you see just like a flipping Then after she turn over again, she just turned that side Follow another arrow, please And let's see how many friends choose which end answer so I'm says all two of them choose be ordered I didn't even start timer. Look at you Okay, three friends says answer should be You One friend text me I guess for France's answer should be Guys just a reminder. Can you please try to share your answers with so on? I Mean yeah, I can still read it for you. No worries guys, but it might take some time one another friend says Be let me just put that one It has been 45 seconds, yeah, you know friends answer let's just make it one minute then we get a start another friend says Be I believe it's six of them. Look at that And this is one minute guys, I don't wanna believe that you'll get this one at least many of you so look Then you turn over the object We are talking about symmetry guys here If I can get the line first There is a symmetry line here Or we can talk about your image whichever you like From that piece. They just turn over You basically are getting The shape just oh, yeah, thanks. Oh, I was about to put that line body and another one should be somewhere here Okay Then as you see guys one piece Little piece is shaded if you look at the shape you will see that that parts. Okay, so I'm you're a little test Thanks so much. You will see that guys. This piece is shaded. All right After that as you see we are not done yet with the flipping or you know rotating the shape I Would say rotate just turn Okay, we gotta do some other Turns For the second part. I am gonna follow this arrow if I can make the line correctly like straight Okay, that's better Like straight Okay, that's better Then guys what we can think of Yeah, we get the work with that another line and we have little tiny line in the middle as you see and we need to Cover look at that shape either top or bottom piece is covered or shaded as you see that bottom or that kind of It for this one right parties shaded because it was on the left So at the end guys This is going to be your final image That little tiny piece Here, I'm just showing you with the green color. I Hope that makes sense Then of course when you look at that You gotta see only answer B makes sense You know that little Gray area we didn't shape us. Of course, it's not the white These parts supposed to be great as now But we show you on the white and black piece. So answer was B any question for this one so far. I Believe you can text us when you I mean you can use chat box when you have a question then we can move on guys All right next one Number two the diagram shows two rectangles whose sides are parallel as you see, okay What's the difference in the length of the parameters of the two? Rectangles this one was from 2017 and your time starts now All right, two friends says, oh, three friends says 24 so far. What about others? All right, four. It has been 45 seconds, guys. They ask us to find the difference, guys. What is the difference? I would focus to get some sort of line, compare. I should compare every single side of smaller and bigger triangle, then look for the difference actually. I am going to explain what I meant. All right, only four friends, come on, we can get one or more, two more. It has been one minute and 20 seconds. One friend would like to chat me, chat and send it to me, I believe, and says, that friend says E as well. Again, please send your answer to Suham, not me. All right, we get five so far. Let's just start, we get already another answer. All right, guys, so when you compare, they say, what's the difference? Of course, we are going to compare to find the difference. I recommend you to do what? Let's try to find some sort of relation between smaller and bigger rectangular shape. Look guys, as you see, I got to use blue color, I believe, yeah, I am just showing you that I will transfer those sides in the smaller triangle to the, I mean, smaller rectangle to the bigger rectangle, guys, here, as you see, I am showing you these pieces. All right, so since we got to find the difference, we can basically cancel them, you know, you don't necessarily try to find the perimeter of each object. You cannot find it anyway, but you know, just say, well, if that piece is three meters, guys, we know that we can transfer that size to here, basically, you know, because that piece is extra, then you compare the smaller and bigger one, and I can transfer that three meters here as well. With the same logic, guys, I can transfer that four meters here, and four is here. Are we done yet? No, because we got to look for those two and three meters as well, that two meters, since these two lines at the top are parallel, that piece also supposed to be two and two. With the same logic, that little piece gives us three meters, the distance between those two lines, that means little extra parts are also going to be three in this case. Well, at the end, you just got to add those numbers, you know. So when you say, I'm going to start from here, the first number we put three plus four plus two plus three plus four plus three plus three plus two, then when you add, you should get 24, or if you are saying that, sir, it's a time test, I really want to double that two times, you should get only one of these, you know, because they are doubled, only one of these, basically. You can get three plus three plus four plus two, you should get 12 times two as well, that's also 24, guys. If there is any question I believe you should ask, you'll ask sometimes, okay. I'm going to move on. And the next one is from 1998, number 12. In a triangle ABC, as you see, there is no triangle, guys, I recommend you to draw a triangle, please, otherwise it's not going to make any sense, anyway, in a triangle ABC, the bisectors of angles ABC and ACB intersect at point D. We know that angle measurement of BDC is 150. What is the measure of BAC? And your time starts now. Okay, after one minute I'm going to draw a triangle, guys, no worries, but I expect you to do by your own first. All right, it has been one minute. And let's give you some sort of triangle. Well, let's these are A, B, C, and I'm gonna say A, B, C, that bisectors something like that. And A, B, C, that one intersect at point D. All right, so we have A, B, C, and this one supposed to be D as you see. And these are the bisector A, B, C, and bisector of A, C, B. I will just put those angles. So for some research, you gotta see that little symbols like that, you know, these represent same angles, or sometimes they're gonna just put little points in the angle, like whatever, you got the point. These are the same angles, like D, C, A, and D, C, B. Anyway, and they say B, D, C is equal to 150, gotcha. And the question is, find B, A, C, guys. We are going to find that shaded angle, please. So far, three friends says answer should be 120. I wonder about other friends, what they are thinking. Okay, four friends says 120. By the way, time flies, guys. It has been more than two minutes. Well, all right, oh, I have also, I have seen getting some chats, okay. One friend says D to me, so I will, that makes, oh, five, even six friends says D. Let's check guys. So look, when you have been, uh, questions like that guys, first things first, you got to do what? Just kind of draw the shape, make vision. Then if you have something, some of the same measurements, I recommend you to use same letter, represent same values here, X and Y. All right. So let's just focus the triangle BDC guys here. We all know that, remember from the one of the previous slides, some of the interior angles in any triangle is 180. That means I can say X plus Y plus 150 is equal to 180 guys. Well, we want to find the sum of X and Y. That means we get, get rid of 150. We should get X plus Y is equal to 30 guys. This is the first thing you need to think of. And as you see, we cannot find either X or Y and we don't need to anyway. Right now, let's just focus another triangle, which is a huge triangle itself. So we have ABC triangle ABC, and I'm going to say this angle is A. In that triangle, A plus 2X plus 2Y is equal to 180. We both know that. Well, here even since X and Y has a coefficient of 2, I can rewrite it as A plus 2X plus Y is equal to 180 guys. This is how we can factor if you like. If we knew X plus Y guys, it would be really easy for us to solve. But wait a minute, we already know X plus Y. It's here, as you see that little tiny X plus Y, it's 30. Just plug them in. We have A plus 2 times 30. It's going to be 180 guys. Well, A plus 60 is equal to 180. Then you got to do what? A equals 120. Any questions so far? Just a reminder guys, for some of the resources, it's going to say, I don't know where to put, but if I can get another triangle really quick. Actually, no, I'm going to just erase that. It's not easy to fit. Okay, so for some of the resources guys, they are going to say that angle of A should be equal to, what was that? 180, let's say this angle, DB, whatever, 180 minus D divided by 2. Was it correct? There has to be some equation like that, or maybe 180 minus A. So it's the formula for this angle. So what we do is the formula goes something like this. So D is equal to 180 plus A all divided by 2. Now, there has to be some sort of formula. If you remember, you can just show us a little bit. Okay, so guys, as you see, if you like to memorize formulas, and you can just use that formula without doing any other work, as you see, you can find either A or D, doesn't matter which one is given. On many cases, they are going to give you one of them, and they will ask you to find another one. Anyway, just plug in the formula and find it, okay? That should be it. Oh, you said this one is plus. If we had those bisectors, I believe, outside, then we should do some sort of minus, am I right? So I don't remember exactly how it was. I only remember how to do algebraic. Anyway, yeah, we have one formula for if the bisector is inside, another formula is bisector is outside, and so forth. Yeah, anyway. No one give us anything on the chat. That means we are good, I believe. Okay. All right, guys, so the next one. Number four from 1999. Oh, gosh, think about more than 25 years ago. The shaded part of the figure to the right has an area of what? Okay, that is your question. You even give you the shape, and your time starts now. Remember, we talked about area of the triangle. It was one over two times base times height. You may need to use that. All right, one friend said answer should be B. It has been 54 seconds. It has been more than two minutes, and only two friends answered. What about others? Guys, let me give you a hint. Look. There are different approaches for this type of problems. If you do not like to find the shaded areas itself, you can work on the whole, find the area whole, and take out those little, small, white pieces, if you like. So far, we have six friends, so we can start. I'm going to explain both of the ways, guys. So look. If you look at the shape, since it's a square, you know, 3xx and 3xx, these are the same sides. If you put that little, tiny diagonal, you will see that the part I shade here and other piece, other gray area, is going to be same, guys, because of the rule of symmetry, as you see. If they are the same, if I find one of the area, I can just double, you know. But here is the tricky part. Where is the base and height of this shape? You may not necessarily see that, but I need you to focus those parts. So if I say, base is that green part for the gray area, I can totally do that. And sometimes, height can be kind of outside of the shape. That's totally fine, too, even if it's uncommon, all right? No worries. Why I specifically choose those sides, guys? Because I have only those sides are given. I cannot find that diagonal or other pieces. No, just work with this part, this side, among other two sides. Anyway, so we know that that part, we call the base, is 3x here, and the height is x. It's here, anyway. So you got to do 3x times x divided by 2. But we have two of those. Then, you know, you can just cancel, and your answer should be 3x squared. I know that some of the other friends might approach differently. And they may think, sir, you know what? I'm going to find the total area, then I'm going to subtract those white parts. Why not? That works, too. So we have 3 plus 1, 4x. We got to say 4x times 4x, which gives us the area, you know, squared. Then we got to do minus other pieces. We have 3x times 3x plus one of that white triangle here is x times 4x divided by 2, you know, x times 4x divided by 2. But we have two of them. I will just double. So you can just work with that. So 4x times 4x is 16x squared minus. So we have 3 times 3 is 9, 9x squared, basically. And x times 4x is just 4x squared. Then you do, guys, 16x squared minus should be 13x squared gives you also 3x squared. Any questions for this one so far? I believe we are good. All right, guys. The next one, which is from 2014. 5 equal rectangles are placed inside a square with a side length of 24 centimeters, as shown in the diagram. What is the area of only one rectangle? And your time starts now. Remember, they are saying length of a square is 24, guys. And these are all equal rectangles. All right, one friend says the answer should be E, so what? And two friends says, let me see. And it has been two minutes guys You can get more than three, come on Others you can guess too, it's fine. I'm not gonna give you any grade, no worries And another friend says E Okay, it has been three minutes. I believe other friends are still working on it. So look Remember from the problem-solving strategies, we gotta exactly know what they are asking It says what's the area of one rectangle? If this is what they are asking finding area of one rectangle here That means guys we need to focus how to find area of rectangle If you remember it was length times width, okay But we have no idea what's length or what's width of those little tiny rectangles But what if we can get some sort of ideas? Let's see. Look As you see if you look at from left to right Or right to left whatever We can look at those pieces. I am just showing you here that one and that one and From left to right we can say how many of those tiny rectangle shapes Just connected to one to another to make it 24, you know Because they fit in the square shape and the one side Actually each side of the square is 24 units or something. So Here is the deal. Let's just focus here from right to left and top to bottom Maybe you can get some sort of sides either length or width As you see I show you by circling these sides are longer sides and we can say these are the lengths But here is a tip we put just one of the Rectangular shape here. Another one is here and in the middle is getting a little tricky guys from here You This piece as you see they took out one of the width From the length since they are five equal same objects, but they add another width here If you look at that, that means While I Kind of show you that little pieces with yellow color. They are the same Since we know that red piece plus yellow piece gives us the whole length Here also Red piece plus yellow piece gives us another length. If you count guys, it's look like we literally Connect one to three rectangles and They're from their lengths basically So we have length plus length plus length gives us three lengths is equal to 24 That means guys length of little tiny rectangle should be 8. This is the first thing you need to do But are we done? No, because we gotta find width as well. How can we find width? Let's just focus with the same formula here. I Just may need to erase this piece. You'll learn how to get length right now guys Let's focus top to bottom piece and try to find some Info you can get here top to bottom But remember we do not necessarily focus on length but we gotta focus on width Here is one of the width here Here is another width here from top to bottom and as you see we have one of the length here and another length here they just Added to each other. So these sides are 8 and 8. We already got that. Well, that means we have already we have only One of these Variables left so total was 24. We can create another algebra equation to find width. We have 1, 2, 2 width Plus We have 8 plus 8 which is 16 and the result should be 24. Well at the end we gotta solve this one by Focusing inverse operations minus 16 both sides I believe you'll get that 2w is equal to 8 and width is going to be 4 centimeters. Okay Then what we can do we will just multiply 4 times 8 gives you 32 square centimeter Any question for this one? All right, ladies and gentlemen, I believe we are good so we're gonna move on to the next one The next question says it was from 2020 2012 my bad in the diagram ABCD is a square M is the midpoint of AD and MN is perpendicular to AC As you see AC is a diagonal by the way what is the ratio of the area of the shaded triangle M and C to the whole square and Your time starts now So far it has been 1 minute and 25 seconds and there is no answer. Look at that. I hope some of our friends are going to give us some sort of answer. Okay, so far one friend says the answer should be C. So far, one friend says D and three friends says C and it has been three minutes. What if I say either of the answers are wrong? As you see, guys, some friend says C, other friend says D. What if one of the groups are wrong? I mean, it's obvious, but do you think which one is that? I expect whoever says D should move on C or those friends says C should move on D. Actually, one of them is correct at least, but not all of them. It's obvious, you know. Oh, someone from C take out their answer and make it D. Okay, interesting. Right now we have 50-50. All right guys, so look. It's obvious that when we use diagonal AC, it divides the whole area by two. You know. Also, it says, when you check every single information, the question, M is the midpoint of AD. That means, I believe you already realize that, the area of DMC and CMA, they are the same because of those same base. Yet if we expect that DM and MA are the base of those two different triangles, that means they are the same area. What else we have here? They say MN is a perpendicular to AC. MN is perpendicular to AC. That means we get little box. We call them 90 degree. Also, we just show as a box case. If that is the case, we know that this is a shape of square. These angles are 45 and 45. These are, and this means that top also 45. And these sides are going to be same. Also, your main goal here guys should be, find some sort of relation between the area of MNC and MNA. Because if you get that relation, that means we literally find the total area of MCA. Then we can make it MDC. Then we can just refer to the whole shape. Anyway, so to be able to find some sort of relation between MNA and MNC, we will either look at the basis, like basis AN and MC, or we're gonna find some other solutions. Look, what if I put another line here? It's not perfect, but yeah, I will do get main idea. Anyway. So since, let me just make exactly from here, actually. Okay, guys. So since we said, I just put that line parallel to the base, that means these are all 90 degrees. And that piece angle AMN were 45, also NM, let's make it K. This angle is also 45. We have 45, 90, another 45, and another little tiny same sides. M and K, let's just make it H, whatever. I'm just putting random letters. So here, guys, if I say the area of that little triangle, the smallest one, which is M and A is A, like capital A, I can represent the area of M and H also with capital letter of A, guys. It's same logic. As you see, we just put one of the parallel line from the midpoint, which has to divide that diagonal by two because of the properties of squares. So AH and HC has to be same measurements. In that case, HC should be double of NH. If area of MNH is A, area of MHC should be 2A, guys, because of their bases. Okay, I will keep explaining, but if you feel you have any question, please ask me or ask us, ask someone. Anyway, because it might be confusing, guys, if you see that concept at first time in your life. Anyway, so then what we can do here, we know that we add those areas, which is area of MAC, we get four As. And remember, we said that area of MAC and MDC are the same. If that piece is total 4A, other piece is also 4A. Well, we can easily say that another piece of the whole square shape is also 4 plus 4, 8A, guys. Then we got to do the measurement, you know. They say, what's the ratio of the area of the shaded triangle? This here, 1A plus 2A is 3A, over the whole, the square. Well, 4 plus 4 is 8, another 8 is 16. We don't need As, it's going to be 3 over 16. In that case, friends, whoever said answer should be C, they might do some little calculation mistakes. Any question for this one, guys? That means we are, okay, gotcha. I am glad to see that you get this. It might be a little off-topic, by the way, guys. Well, I got that clear all. Then go to the next one. All right, from 2003. Six consecutive points were indicated on a number line in this order, A, B, C, D, E, and F. Regardless of the location of these points, if only AD is equal to CF and BD is equal to DF, then the following equation is true. Which one? And your time starts now. All right, so far, one friend says answer should be D. Other friend says, no, actually answer should be E. So far, two friends says D versus one friend says E. all right so far four of our friends says answer should be d and none of the other friends says answer should be another okay let's see guys so first thing first we need some sort of straight line as you see guys then we got six consecutive points are going to be indicated on this number like a let's say b c d e and f it should be enough a b c d e f gotcha let's see regardless of the location of these points if only ad is equal to c if i change the color it should make more sense to you um it says ad is equal to cf ad here as you see and cf is this piece guys also they give us another equation and they say wait a minute also bd is equal to df where is bd bd is here this part oh yeah so you put that and edf should be this piece oh yeah gotcha but let me just try to put this one too kind of i'm kind of trying to use straight lines so i'm not against you're helping me thanks so much you're doing great okay guys at the end they then they said which of the following equation is true they say ab is equal to bc well do you think we have any information about ab and bc not necessarily at least when you look at those lines they say bc is equal to de bc is here let me change the color first bc is here and d is here again guys not necessarily any information they say bd is equal to ef bd those two and ef only one unit most likely wrong anyway this says ab is equal to cd since many friends says that one should be answer let's just check this look we have given ab and we have given cd look at the lines guys we have that i believe i use i don't remember which color i use anyway this color like a bright black color black color and minus pink color here piece as you see we have to find ab we literally have ad minus bc we have only that piece left over here as you see and here guys we when we compare cf and df we have only this another green color green part left and they said ab is equal to cd well since those measurements are already given and they are equal that means leftover pieces which is green pieces has to be equal as well that is the reason guys answer should be d you don't necessarily check it any question for this one all right i believe we are good guys and i enjoy our lesson i hope you enjoy as well and i'm gonna see you next time bye

2D geometry

regular hexagon

intersecting points

geometric concepts

Pythagorean theorem

sum properties of angles

geometric relationships

problem-solving

area and perimeter

collaborative learning