Welcome to another webinar session. Today, we are going to talk about some properties of 2D geometry, and we are going to go over some of the MatConc related questions. So this is your warm-up for 2013. And question number one, let's see how many of our friends get the answer for this one. Thank you. Bye. All right more and more friends are joining. Welcome guys this is our first question. If you have some ideas about how to solve it please direct message to Soham so he's gonna let me know. Suhab, do we get any answer so far or not yet? Not yet. Okay, I will have the answer later. Alright guys. Bye-bye. One man says D. Okay, one friend says the answer should be D. Alright, anyone else? All right guys let's start. Since we have given big equilateral triangle and it says the lines are parallel to the sides and divide the sides into three equal parts. That means if I say these pieces are X each of them are supposed to be X because of the properties of equilateral triangle. All right so let's just divide this part into two pieces so as you know guys from any equilateral triangle each angle each interior angle should be 60 here and those sides are the same that means we are going to have the same angles here 120 180 minus 60 gives you 120 120 divided by 2 also gives us 60. Wow that means we have every single interior angles are 60. Well that says that we are going to have same side length this piece little piece is also X. Since it is so small let me just show you like that this part is also X. Okay with the same logic guys if we kinda separate those shaded parts into two you will see that these pieces are going to be X as well and since we have all of those parallel sides these are all parallel and we gotta get same 60 degree as an interior angles that means every single side of those smaller triangles has to be same guys. Well it has been said we are going to have 1 2 3 4 5 6 7 8 9 of those little small triangles. We have 9 triangles I am NOT talking about the bigger one as you see I am talking about those small ones 9 triangles which the total area area of 9 units square. Well if it is 9 units square that means guys each of them one triangle each of them is going to be 1 area of each of them is 1 unit square. Well that gives us what's the area of the shaded part you gotta look for or just count them we have 1 2 3 4 5 6. 6 of them is going to be equal to 6 guys. I hope that makes sense as you see we use properties of equilateral triangles similarity parallel sides and area okay as you see even one triangle or just one geometric question you need to remember four different topics related to geometry so please make sure when you study geometry just you know go over the previous topics as well. Any question about this one? That means we are good. We can continue to the next one real quick. Alright as you see we are going to talk about supplemental 2D geometry a little bit deeper of the 2D geometry. Some of the rules about this topic. This topic involves all problems with shapes or objects that occur in flat plane basically 2D geometry. You should be familiar with variety of 2D shapes. Yes. Problems can include line segments and segments and right rays. This can be parallel intersecting. Mad Kangaroo may ask you to use your knowledge about angles formed by these objects. A right angle measure is 90 straight lines measure 180. A complete angle which is kind of circle also gives us 360. The sum of angles in a triangle is 180. You can use equal angles and side of equilateral and isosceles triangles to help solve many different problems. Right triangles have the 90 degree angle. That's the reason we call them right triangle. The length of one side have special relationships which is a square plus b square equals c square but if you confuse guys about which is what is a what's b what's c you can also say those 90 degree sides we call them leg actually and this is hypotenuse you can say leg square plus another leg square it doesn't necessarily mean they are the same measure you know is equal to hypotenuse okay if there is any question I will be you are just gonna type so I mean you're gonna ask so how many in the chat so I will you know go back I am gonna move on guys because I wanna cover as many as problems we can solve. 2D shapes exist in a flat field or space also called plane. Special notation exists on described xy plane here as x represents horizontal y represents vertical axis here and x increase while we go from left to right y x y x y we also increase while we go to from bottom to top. Point segments lines trace and 2D curves and 2D shapes can be contained in a plane we can you know form them. Every point in the plane can be described by it is xy coordinate plane. The xy plane is divided into four quadrants based on the positive and negative coordinates as you see. We can just say first second third and fourth quadrant. On the first quadrant we have both x and y positive guys. The second one is x has negative numbers like a negative positive. The third one we have both negative numbers that means x and y values are going to be negative. I am specifically telling you because we have questions related to that specific coordinate plane. And four represents x has positive values and y negative values guys. Again I believe you asked if you have any questions about those. So some of the vocabulary words concurrent means they are exactly the same two objects the same shape and dimensions. Similar two or more objects that have same shape and angle but differ by scale factor. Scale factor means guys some sort of integer you are going to multiply. It can be fraction as well. You are going to multiply one object to make it bigger or smaller. So as you see if I say those two triangles are concurrent that means they are same. I did my best to you know make them look like same. But if they are similar guys one of the triangles can be like that. Equilateral I mean isosceles triangle and another one can be like that. As you see maybe you have like side length x and 3 axis here. The scale factor is 3 here because you multiply each side by 3. Anyway radius of a circle is a distance from the center to the border of the circle basically. Diameter is 2 times the radius or the length of a segment crossing the circle through its center. Tangent objects intersect at exactly one point. Like those two lines intersect here and we call them this point tangent. Okay we can move on to the next. In solving that kangaroo 2d problems you may find it's helpful to draw situations that are described when no figure is represented. Yes guys sometimes you gotta see like whole paragraph of the question and there is no shape. I recommend you that 99% you are gonna benefit if you just draw something. I know you may or may not be a good artist but it's totally fine. No worries guys. Draw something. It's gonna give you some ideas. Think of alternate ways to the situation could be drawn. Drawing additional points and line segments like for example we have given triangle but the question might relate exterior angles. In that case you can just kind extend those lines you know to be able to think about exterior angles as well okay. Observe how shapes intersect and meet. Imagine shapes move flipped or rotated. Use algebra to describe or solve some of the problems. You know in almost 99% we are gonna use other equations. Use logic to determine possible outcomes. Then you can eliminate some of the wrong outcomes. Alright so we have your first question from 1998 and number 4 guys. Let me start your timer. Let's see how many friends are going to be able to find that one in less than one minute. all right guys it's one minute I hope some of the friends were able to find something two people who say a all right and then say a okay we have three of them right now yes one person says we all right friends so even we have a shape I try to make it you know easy to understand for you we are going to fold along here so that c overlaps with b as you see we are looking for that kind of purplish color they are asking us this one well three friends says 1.5 my understanding those friends find if they ask us to find that green piece if the question you're saying that that's corner a fold along here and this piece kind of fall apart that would be you know 1.5 but answer is not 1.5 I'm sorry guys whoever says a please change your answer really quick or at least check your answer please people saying b how many sorry you're one person who says b and then one of the a people switch to b okay so how many total b right I'm confused man sorry three okay two in the a okay we have two says still a three says b all right guys let's start here for those case of those type of questions and these cases you need to use similarity guys you can say but sir I never ever see the you know question relates similarity yes you haven't seen similar the word of similar but they are still similar what I meant here guys let's say this abc and we have de here first of all if the point c overlaps and when we fault comes to here this piece cd has to be cool to db guys because it's exactly false by half you know I mean I believe you fold paper a couple of times in your life anyway so with that has been said also this has to be parallel because this 90 degree also says this part is also 90 degree guys when you fold that okay to be able to you know fold perfectly anyway we have 99 degrees when you look at the triangles angles of the triangles c ed and c ab we have the same angle at the top they share since ed is parallel to ab also that little red angle here also equal to here that's another red angle that means guys we have to say that c ed triangle cd is similar to triangle c ab if I just put you look okay that's meant guys well if we have similar triangles we can just find the relation between sides since we know that cd over cb cd over cb here is equal to one over two with the same logic we can say ed like here last to ed over ab well we all know that ab is four that give us one over two is equal to let's just make x x over four I believe you all figure out that x should be two or you can just do you know cross multiplication butterfly whatever method you want like that okay x equals two guys two times x is going to give us one times four and x equals two and answer was b guys as any question do we have here as to no question has been asked then I'm gonna move on then all right all right what about number two guys this question from 2007 and question number 12 to be able to solve this one please read the question carefully if you are careful enough you gotta figure out how to solve this one it's not that difficult but you gotta look for the properties really carefully anyway let's just start timer Bye-bye. Wait guys, it's been more than one minute. I still haven't get any answer yet. I don't know why. Nothing yet so far. I am sad man, what are you doing? I literally tell you that little, actually huge hint, they are squares. What does square means? What's the main property you need to use when you work with squares? Nothing yet? At least one answer? Maybe one friend just guess? Okay, one friend said B. Okay, guys, another reminder, please send your message to Sogamsol, he's going to be able to check easily compared to me because sometimes I'm gonna ask, you know, just keep texting. While I am teaching, I might not be able to check the chat all the time, so... Okay, one friend says answer should be B. So, two people, yeah, two people say B. Okay, another one says A. One of them says A. All right, let's see, we have B versus A, and C, D, E still get nothing. All right, guys, let's check. As you remember, I said, think about, you know, properties of squares. If I say length of that first square is little tiny A, and I'm going to keep going with the same logic, this is B. And the third square, length of third square is C, D, this one is E, the source one, I'm not gonna write every single signs, and the last one is F. Okay, look, you're gonna say, but sir, you give us like literally a lot of unknown values, like what? Guys, we have given A, B equals 24. If you look at the shape carefully, A, B represents A plus B plus C plus plus E plus F is equal to 24. But you may still ask, sir, you said that, but we have only one equation, what about that A, A1, A2, A3, whatever B is, what is that piece, I can just show you a different color, how can we find that little piece of line, so basically, total of the line segments, I am showing you the pink color, how are you gonna find that? Guys, if you have no idea still, you can just keep following. Let's start from A to A1, we have one of A, as you see, plus A1 to A2, another A, A2 to A3, we have A plus B, as you see. We got another person saying, people are saying C now. Oh, everyone says C, interesting, okay. We got two in C. Okay, two C, let's see. Okay, and then we just keep going, we were here, guys, I put another B here, as you see, and I have B plus C. Plus C plus C plus D, I just, you know, look at those pink piece and just keep adding sides, okay. oh it takes time man look instead of having a a a plus b b whatever as you see guys we have three a's plus three b plus three c then we have three d three e and three f wait instead of putting them like as a three a plus three b plus three c can i make them uh use factoring rules can i just make like three a plus b plus c plus d plus e plus f can i rewrite them like that guys algebraically it should be fine if you are able to follow with me you can say yes okay so right now as you see even they give us that little pink weird pathway we gotta find the total length we have three times something do you think we know a plus b plus c plus d plus e plus f is yes we know that but as you see from the first part the question says segment ab is 24 which is a plus b plus c plus d plus e plus f well you can just plug them in here 24 and as you see guys your answer should be three plus i mean three times 24 which is 72 all right some friends found other answers at the beginning maybe they just do some calculation mistakes i don't know guys but this is how we are gonna solve this as as i told you before please make sure either draw the shape or put the properties on the shape guys square means all of the sides are the same then put those same letters at least then you gotta figure out something you gotta realize something but if you don't do that you will get nothing so if you get a chance to convert geometry to algebra it's gonna be most likely easier to solve any question about number two well that means no okay let's move on number three then all right another 2d geometric equation with the shape from 2015 number 12 let's see who's gonna get what um All right, guys, it has been one minute. I believe someone got the answer, so I'm going to tell them already. Someone says C. All right, we have one answer, says C. All right, guys, let me give you a hint. You will use or you can use similarity, properties of similarity. What does similar mean? How do you apply similarity rules to geometric shapes? It has been almost two minutes. Since we have been talking about similarity guys, I recommend you to check those triangles I'm showing you with the kind of yellowish color. We got two more to say C. All right, look at that. She got three votes. What about others? Anyone says answer should be D. What about A? All right guys, when you focus those triangles, let's just start with the properties. Those angles we call them vertical and they are the same angles. Since we had the squares, these are little box which is 90 degree. And those angles I'm just showing the leftover angles should be same as well because other two angles are same anyway. Okay. As you know, because we have a square shape this side and this side are the same. With the same logic guys, you can say those triangles are similar. Also, let's just say A, B, C, D, E. I would say triangle A, B, C is similar to triangle. You got to start from with the corresponding sides. Remember that vertices A match with D. I'm going to start with D. D is the middle. It should be 90 degree which is C. D, C. Oh my bad, D, E, C. My bad. This is going to be E guys. D, E, C. All right. Well, first thing first, we say they should be similar. But, there is but. If we know that those sides are also even equal to each other because it says each square has side with a length of one. Remember guys, similarity means when you divide corresponding sides, you get some sort of numbers, fraction, decimal, whole number, whatever. But in many cases, it's not going to be one. But in that case, corresponding sides, which is A, B over D, E is equal to one. That means beyond being similar, those triangles are the same guys. We have same exact shapes. Well, if we know that they are the same shape guys, can I just take out that shaded area and put it here? You know, I'm just showing with the green color. Well, as you see, I transfer or kind of rotate triangle D, E, C and put it exactly on A, B, C. At the end guys, you should get the shape of one whole square, which has the side length of one. Well, you know, the question becomes really easy right now. Area of square is one times one, you know, one times square is just one. That's it. Any question for this one? I mean, look, it seems so easy to me. I hope you feel same as well. But remember, I warned you before, we got in similarity a lot. This is what we are doing as you see. Please do not just look at the question and look at the shape and says, no, I cannot solve it. No, man. You already lose it before you even start to fight. Okay. Anyway, since Soham says nothing, that means I believe we can move on to next one, number four from 2021 and question 18. Remember guys, we have given some of the properties which says here, those, there is relation between those areas P and Q, you know, and those side lengths dimensions are also important. You need to find the value of x, please. Remember, we are going to use algebra. Please do that. Try to create some algebraic equation and you are good. Okay, I gave you enough hints. So your time starts now. Thank you. All right, it has been one minute, do we have an answer? Okay, I'm gonna wait more. There is one person who says D. Alright, we have D. D got one vote so far. It has been 2 minutes 15 seconds, guys. Look, let me start from the figure on the left being folded to make its figure on the right, guys. So whatever side length and stuff you see, I would just plug them in on the right, you know. So this part or width should be 4 here. I'm just gonna put it here. And this piece, think about it, in any rectangular shape, width are going to be the same on the right or left, whatever. From left part and right part, this piece also is possible for this. So since that part is given X, let's just make that little side length. I am gonna show you by red color. Let's say this piece is Y. You may say, sir, why do you give that piece Y? Well, because I need to put one of the other guys to be able to compare side lengths of X and Y because as you see, we have given, they compare the areas, P is equal to 2Q. Well, when we find area of P, we can say 4 times X, you know, length times width, width times length, that same thing. But area of Q says 4Y. They say P is equal to 2 times Q. Well, let's just plug them in. P is 4X, is equal to 2 times 4 times Y. As you look, I just plug them in. Well, those 4s are gonna cancel. It gives us X equals 2Y here. I can just put here 2Y. What else I can think of? Well, as we know, guys, we have, this length is 13. Am I right? Think about 13 as well. Which piece is 13? Look, I am gonna show that part. Let me change the colors too. Maybe that one, ice blue, look like, oh, it's gonna look like that. Okay. Look at that ice blue color, guys. I am gonna continue from here. Since from that corner, we fold the shape, fold the little piece of paper, that ice blue color is equal to, basically, length of the rectangular shape. It's going to be 13. You know, I'm just gonna show you, it's same color, meaning same color side represents same length, okay? Anyway. So then we can say, guys, this piece was X or just 2Y, plus we have, as you see, 4 and 4, this piece is also 4 because of the little rectangular shape, remember properties of rectangles, plus another Y here. The total was 13, as you see. Well, 2Y plus 1Y is 3Y, plus 4 gives us 13, as you know. You gotta solve that. If I ask you what's going to be first step, I believe you gotta say, cancel 4. Yes, I am gonna do that, guys. So 3Y is equal to 9, which is 13 minus 4. And what's next step, guys? You may say, divide by 3 because we need only one Y, not just 3Y. Okay. That means, guys, it gives us Y equals 3. So you may say, but, sir, there is no answer that represents 3. I know. So because they ask us to find X, X is equal to 2Y, remember? That has been said. 2 times 3 is 6, guys. Answer was 6. I don't know if any of you were able to find that or, you know, you were working, but answer was 6, guys. Any question about this one? One friend says 6.5. I assume they just do calculation mistakes. Okay, I am gonna warn them. All right. Next, from 2013 and question number 16, I told you we are gonna have the question related to coordinate geometry. Think about those quadrants. It's quadrant 4. And anyway, I am gonna just start timer, guys. All right, it has been one minute guys, as you see I give you some hints by either underlining or circling the key parts You know one who says a and one who says B Okay, a got one vote and B got one vote so far The B person changed his answer to A. Oh, okay, minus one plus one, all right What about others? I wonder if someone thinks it's E, it depends on the rectangle, you know The third person says A It has been two minutes, guys. All right, let's start. So look, for those, if you like, you can just use some of the properties of the coordinate plane and the rectangle of the coordinate plane, basically. Or if I were you guys, I would just plug the numbers. I mean, why not? They say a, b, c, d are all integers. Okay. Let's just start here. I can make it this point, point d. 1, 1. You know. 1, negative 1, like that. x plus y is negative, you know. So... For this one, let's just say this is... I'm just making up. For... x is gonna still same. Let's just say x is still 1 and y negative 4. Here, y is going to be same, you know. Let's just say... x is 2. y negative 4. And for this one... It has to be x is 2 and y is negative 4. Okay. 2, negative 4. Then we just gotta do y over x for every single of them, right? So... Point a gives us whatever. b gives us something. c and d. All right. Point a says... We have y over x, which is negative 4 over 1. Which is negative 4, my bad. Point b says... Negative 4 over 2. Which is negative 2. Point c says negative 1 over 2. And d says negative 1 over 1. As you see, guys, they are discussing about the least value, which is going to be negative 4. You may still say, sir, how do you know this is the right order? I am saying, I don't know. This has to be coordinates of the numbers. But, again, if you use the properties, as you know, from point a and point d, x values has to be same. And from d to a, y values has to decrease, guys. Here you are dividing different numbers by same number. For one case, for point a, we divide negative 4 by 1. For another case, which is d, you divide negative 1 by 1. As you see, of course, the value of a has to be less than the value of d anyway. You can use the same logic for c and d, or when you compare c and d, because when you check c and d, you will have the same y value, but different x values. From left to right, x increase, you know. And also from bottom to top, y increases as well. Just apply the properties, plug the points accordingly, and it doesn't matter which number you choose. If you follow the properties, you will get the same answer as a. Any questions for this one? All right, I believe we are good. Then I will move on to the next one, guys. Number 6. And let's start your timer. I am heading out bye bye yeah thanks so much have a nice holiday and see you in the next webinar all right thanks so much yes we will take care man all right guys he had to leave so last about nine ten whatever minutes you can send your message to me i will try to make it easy for us and it has been 1 minute and 16 seconds guys any of our friends are able to say something okay i got one message let's see okay one friend says it should be c what about others she got one vote so far all right another friend says answer should be c she has two versus others got nothing he's almost two minutes guys we're about to get two minutes all right i have one more friend that friend also says c look at that maybe in that case we should think answer should be c let's see guys all right look these are the some of the classical questions you gotta see in not necessarily in many many different competitions guys if i can use the properties of triangles you can say what there is no triangle here then we may we can make it one you know if there is no triangle then we will have our own train okay something like that all right yes i try to make it as perfect as possible but you know it's not easy that case so we have right triangle here as you see guys okay from the properties of given circles since these are the center here and here this piece is also 17 that piece of 70 i mean 9 not 59 we are working with the radius remember all right as you see we do not know what is that piece let me just put here a b c as you know guys we don't know what's ac because it's not coming from here tangent because i try to make it right triangle to be able to find you know some sites anyway what else they say find length of a rectangle given that it is width is 50 okay we have given 50 already guys if this piece is 50 you know from that part this part is also 17 again i'm just using properties of circles that piece is also 9 and that piece is x we all know that we have straight line here 17 plus x plus 9 gives us 50 guys well when we add them i recommend it at the like terms for 17 plus 9 gives us 26 plus x is 50 i assume you know the next step which is canceling 26 x is 24 guys here all right so guys as you know we have one of the right triangle remember i mentioned at the beginning of the class when we have right triangle we can use the Pythagorean square square leg square plus i mean the Pythagorean theorem leg square plus leg square gives us hyperbolic so we can see that a c and a b are the legs i couldn't make this one kind of box look like right angle you know i know you are smart if you get out okay let's say this is side length b you know something like that okay we can say b square plus 24 square gives us 26 square okay so guys without doing any calculation i will just answer that you don't need to calculate all the time because this triangle look like the double of 5 12 13 special right triangle you know there are some special numbers like 3 4 5 5 12 13 uh what are 7 24 25 and so far also 8 15 17 anyway these are some of them so we know that when we double that we still can do that we have 10 24 26 well i am not gonna just use guys Pythagorean term to waste time i will say that b is equal to 10 years okay since b is 10 we just use special you know triangles properties since b is 10 guys i can just plug it here and i am looking for that question mark we all know that this part is b this piece is 9 i am using again the properties of the radius when i extend this piece this piece is going to be from a to bottom and another 17 well then you add 17 and 10 plus 9 i am looking for that vertical piece which is a question mark you should get something 10 plus 17 is 27 plus 9 is 36 guys whoever says 36 they get right thanks so much all right i believe we can check one more question we still have some time any question about this one please send me message really quick all right i believe we are good and the next one please check question number seven this one was 2011 question 29 let's start the timer guys All right. We don't have too much of time left, guys. It has been one and a half minutes. I am going to start, guys. All right. Look. As you know from geometry questions, guys, we need to work with shapes. And for those cases, if they ask you to cut the shape and put it together in another shape, you know, of course, you got the rotations. I would just use the coloring method. And let's say this piece is kind of yellowish color here, here. And since y is going to come back here as a smaller triangle, guys, I can make these pieces here, here. I hope I can show you how they are going to overlap when I just use those colors. And this piece goes here, guys. OK, anyway, we still have the same shape. The leftover piece, you know, that piece goes here. You don't need to rotate or try. You only translate this. You don't have to rotate or reflect or whatever. Anyway, let's just get the red color. And here, guys, from this piece, oh, I forgot to. Oh, I shouldn't do that. This part was here, guys. OK, that's better. And as you see from the properties of the rectangular shape, 11 plus 13 give us the bottom, which is 24, guys, OK? Well, we know that this piece, kind of orange color, is 24. Wait a minute, you've got to say, sir, if this piece is 24 and that little tiny red piece, I am just showing you here, is 13, this part. When I have, when I add 13 plus 24, guys, do you think I am going to get pink color, pink side? Yes, guys, this is exactly what I am doing here. Hopefully, you can see the colors. Well, which is x here, guys, that means x is equal to 13 plus 24, guys, which is 37. As you see, the question was 29. It is one of the hardest questions from, oh, 2011. Can you believe that? But we just solved it really quick. Anyway, if you know how to work with problems, you can solve them easily. Any question for this one, guys? Or any questions about other questions, like general? OK, guys, well, this one was our last session for this set. And I wish all of you good luck with your MatFanguru journey or another mat competition journey. And take care, please, guys. Bye.

2D geometry

equilateral triangles

sum of angles

parallel lines

geometric principles

algebraic methods

X-Y plane

similarity and congruence

isosceles triangles

geometry problems