CAT Coaching – Understanding Geometry – Part 1
So let me start over with. I’ll take the first session with geometry today. This has to be one of the easiest areas. There is a clear reason for that, because geometry is nothing but numbers and diagrams. At its lowest level you can say geometry is nothing but numbers and diagrams. So now out of the four formats which we’ll be discussing all the topics, i.e. diagrams, words, variables and numbers. Most of you will find this, or if you can understand concepts and diagrams, that will, that’s when it will become the easiest. The effort required from your side will become minimum, if you can learn concepts and diagrams. So geometry is all about diagrams anyway. That too diagrams which are very easy to predict. Nowhere close to the level of thinking which we can have. So this is an easy section. It is already an easy section maybe for a lot of you, but it’s going to become very, very easy that you’ll stop preparing for it. Very soon. And geometry and all I can, with full confidence I can tell, please don’t practice. There is nothing to practice in geometry, what you’re going to, and that too maybe if you want I can actually show you the last 50 questions in geometry. If I do that on day 1, I’m very sure you won’t attend geometry sessions. But that we’ll do only in the end because first I want you to reach a level where you can start expecting or predicting, predicting questions in geometry. And let me also tell you that this is actually out of, numbers, variables and all, this is actually my weakest area. That’s when you’ll find maximum amount of methods, or to ask my weak, it was a weak area for me before. But once I understood that it’s all about geometry is circle to or line to a circle, and everything in between. And in one sentence plain geometry is line to a circle and everything in between. And what we are going to do today is, let me, I am going to try a slightly different approach in terms of understanding properties. The ones that you have memorized long back. The ones which you think you need to memorize again. So what will be the number of properties? You might tell there are 24 properties on circles, 36, 48, depending on which book you’ve been following, or which institute you’ve been following. I’ll make sure that you’ll stop, you’ll stop following all of them, after this session. So leave the complete session to me. Just sit here. 50% concentration is more than enough. You’ll end up learning whether you want it or not. You’ll end up learning whether you want it or not, because, I’m telling with so much confidence because it’s nothing at our level – 8th standard, 9th standard, 10th standard geometry. Maybe something where we have wasted 60 hours in our, of our time in schools and maybe another 120 hours at home. And you’ll soon realize, in 6 hours you’ll reach a higher level. So, it’s an easy session and interesting session because it’s all about diagrams. Now, what is that we’ll, you’ll end up seeing in geometry is line, 3 of them together, 4 of them together, infinite number of them together. Right? And maybe somewhere in between you’ll find hexagons and octagons and etc., etc. But all of them are, the most important thing is these are all diagrams which you can predict. They are not going to ask questions based on diagrams which you can write in words. This diagram you can’t write in words because you don’t know the name. So, and normally the way it is, is weight is, if you see previous papers it varies from 10 percentage to 20. 10-20% of the questions, right? This is what you normally call geometry. You’ll soon realize that geometry is much more than that. And much easier than that. So, why I am telling like this is because effort required from your side is very minimum. So let, I’ll just take you through first 10-15 minutes, the same old geometry you have memorized before. Then we will see why you should not do that now onwards.
So, let’s start over with, geometry starts with a point. Plain geometry right? Because I can only do plain geometry on this screen. So as of now I’ll only talk about plain geometry. Whatever I can do on this screen. So it starts with a point. Lot of them together is a line. Right? You can interpret this as one point becoming many. One point becoming many. It’s just a 30 second overview of what we are going to do. Now lot of lines together, now lines again can be expressed using variables, coordinates. Two points are enough to define a line right? Two points are enough to define a line. But if you keep on adding lines, you’ll start getting polygons, triangles and then further polygons, then further polygons, then further polygons. So that’s the last one. Last one being, you can say it’s a circle. First one we normally take it as a triangle. So triangle to a circle also it’s the same thing, one becoming many. One becoming many. Now, adjusting the other way around, the circle, if you cut the circle, straighten this out, you get a line. So line in a way can be taken as a circle with infinite radius and circle can be taken as a polygon with infinite lines. So this is why if you actually think about it, circle is polygon with infinite lines or infinite sides. So if you have a line, if length of this line is fixed and you try making triangle, or a square, or a circle, which one will have maximum area? Using the same line, which one will have maximum area? Circle, followed by this, followed by this. So let that be the easiest CAT question which you will ever see. That’s an actual CAT question. That’s an actual question. Arrange them in the order of their area, where the length of this line is fixed. So more the number of sides, more the number of sides, area will be maximum.
Now, let’s start up with points and then lines and then polygons, up to a circle. That’s what we will cover in, most of it you’ll understand today itself. We’ll solve lot more questions going forward, in class itself. There won’t be nothing much left for practice, as such. For practice you can start solving them. So, now, about a line. What is a line? A line can be defined by, so more than line, I wanted to think at the point level, because that is actually at the lowest level. And that’s what is going to make a difference when you solve questions also. Don’t let properties based on circles and lines. Learn properties based on circles and points. Then you will solve it faster. So, triangles, we will look at both the first one and the last one, today. Triangles and circles. And in between ones anyway, the methods will work. So, now what is a straight line? A straight line can be defined using just two points right? If you have two points, one line is defined. So, two points are used to define a line, or you can say the line can also be defined using, we use the coordinate axis, line can be defined using the slope right? You consider tan Θ where Θ is with positive direction of the x-axis you can define a line. Based on that you can say that lines are either like this or like this. One is where, this is with, as in, when you move towards right, here the value is increasing, and here it is decreasing, so this is positive, this is negative, in terms of slope. Now, lines can also be written. Rather than visualizing the line, you can write the line right? You can write the, you can write any line. So if I just write 3x + 2y = 6, this means it’s a, this is a straight line. Now, there is a specific straight line where the two points are like, you can write like this also. You divide by 6, you can write it as x by 3 plus y by 2 equal to 1. So I am starting with straight lines. So this is a straight line. x by, sorry, x by 2, y by 3 equal to 1. That is nothing but divided by 6. Now this kind of a line, it means x-intercept is 2, y-intercept is 3. Or in other words, you take x-intercept of, here the x-intercept is 2, so this is the point which I am talking about, where it’s 2. And y-intercept is 3. So this is the point which I am talking about. So, and again we know that once we get two points that means line is defined. So, the meaning of this line, in words now, variables here, in a way you can visualize it in a diagram. If you want to write this in words, that what at a level which all of us can relate to? Now this line it means that suppose, if somebody is at this point, this can be written as, falling 3 and moving 2 to the right. That’s the meaning of this line. Falling 3, moving 2 to the, when you are, when you drop something by the time its falls 3 distances, unit distances, it will also move 2 to the right. That’s the meaning of this line. the line is defined by that right? This is actually at a very low level. I am talking coordinate geometry, but something that you can teach anyone. So, if it’s the other way around, if it is the other way around, you can have like, instead of dropping 3 here its, drop 3, dropping 3 and moving 2 to the right. You can also have lines where it’s the other way around. Climbing 3, moving 2 to the right. That line will be something like this. So, and that’s all from, you can directly write that from here right? You can write it from here. if you see something like this, if you see something like this, it means falling 3, falling 3 and moving 2 to the right. So that’s the meaning of this line. Now, so straight lines, obviously we look at this in coordinate geometry also we will have a short session. Now coordinate geometry is not different right? You define by using coordinates, using you know, using both the coordinate axis. That’s coordinate geometry. There’s nothing different over there. Geometry in a different format is coordinate geometry right? All of you know that. So, even if you don’t know anything, there is nothing to worry because what we have done is anyway, mostly you’ll consider waste very soon. In schools or colleges, whatever you did before. Even if, whether you are from IIT, or whether you are from a non-math background. You’ll learn how to learn better. That’s what we are going to do. You’re not learning anything. You’ll learn how to learn better. That’s the only thing which I’ll do. So this straight lines, if you take lot of them together you’ll start getting polygons. So 3 of them will define, 3 points will define a triangle right? 3 points will define a triangle. Infinite number of them, if you put them together, that’s a circle which can be defined by this center and this distance. Right? Center and this distance. By moving this point with the same distance, that’s a circle. You can define it by using this word also. What are the properties, the important ones which you need to understand, if your aim is to just solve questions. Then that makes it very, very easy. But we will anyway look at it from an overall point of view.
So, in a triangle, in a triangle and this is, this part some of you might find boring because you’ve seen this enough times. There are different possible, you can define lines and points based on, now, again I don’t want you to note down all these things. They are in all the books and including the material which we are giving it to you. If you think you need to go through, please go through 5-10 minutes, that’s enough. Altitudes, medians, perpendicular bisectors and angle bisectors. These are different lines in a triangle. If you are familiar with these names it will help you to solve questions based on these lines. If you are not familiar you’ll have to leave it faster than everyone else. So, point of intersection of altitudes is what we call orthocenter, all of you know this, it’s just revision of high school geometry. Medians is centroid. Point of intersection of perpendicular bisector is circumcenter. Point of intersection of angle bisectors is incenter. Now it’s not that you’ll get a lot of direction questions based on all these terms, but you can always get questions based on find the in radius of a particular triangle. Find the in radius of a right angle triangle. So, point of intersection of altitudes is orthocenter, medians is centroid, perpendicular bisector is circumcenter, angle bisectors is incenter. Now all these things are almost self-explanatory. What all these things mean. So what happens is, in a triangle, if you, the line which will divide the opposite, this is the median, because these two are equal. Now, this is the altitude, i.e. a perpendicular dropped from this vertex. This is an altitude. Perpendicular bisectors, at the midpoint if you drop a perpendicular, that’s the perpendicular bisector. And this is an angle bisector. So point of intersection is all these points. Apart from that, different methods of finding the area. What are the different area? In variables you can write it as, again, all of you know these things: half b h, or root of s into s minus a, s minus b, s minus c, where s is the semi perimeter, where a, b, c are the 3 sides. Or it can be, if circumradius if it’s given directly, then it’s abc by 4R, where this R is the circumradius. Or you can also write r into s, where this r is the inradius. Circumradius, inradius. Again, there are so many other ways of finding the area based on some special triangles and all. We’ll look at them while solving questions, all of you know that also. So, basic things I am just taking you through. Now, in a circle in or equilateral triangle what happens is, now that’s a special case where all sides are equal. So what will happen is all these four lines will be the same. Now this particular line will be the angle bisector, will be the perpendicular bisector, will be the centroid, sorry, will be the median as well as the altitude. So, in an equilateral triangle this point will be orthocenter, centroid, circumcenter and incenter, all of them. Now one more important ratio here, which again, it’s just revision, centroid will divide median in the ration 2 is to 1. Centroid will divide median in the ratio 2 is to 1. So this is the circumradius, and this is the inradius. So circumradius to inradius, the ratio, in any, this is a point which keep in mind, circumradius to inradius the ratio will be 2 by 1 in an equilateral triangle. So that is, circumradius if it’s two third of the height, as you can see it’s two third of the height, inradius will be one third of the height. Height anyway will be √3 by 2 a, and area will be √3 by 4 a2. Height will be √3 by 2 a, where a is the side, and area will be √3 by 4 a2. This is an equilateral triangle where all sides are equal. Now in a right angle triangle, that’s again one more special triangle, in a right angle triangle, two ratios, two important ratios, from problem solving point of view, in a right angle triangle, in a 45-45-90 triangle sides will be in the ratio 1 is to 1 is to √2. In a 30-60-90 triangle, sides will be in the ratio 1 : √3 : 2. So two important right angle triangles which you will end up using in almost every other question. So, 1 is to 1 is to √2, and 1 is to √3 is to 2. Opposite 30 its 1, opposite 60 its √3, opposite 90 its 2. And here it’s 1, 1 and √2. So, plus right angle triangles are the ones which we normally express using variables as x2 + y2 = z2. And in a right angle triangle, what will happen is if you consider, in a right angle triangle, this is the circumcircle, circumradius, as you can see here in this diagram that will be half of the hypotenuse. In a right angle triangle, circumradius will be half of the hypotenuse. The circumcircle is nothing but which will pass through all the three points. And this is nothing but the circumradius which is half of the hypotenuse.
Okay, now basic properties on circles if you consider, I am only discussing basic properties just now because all the, everything else you can connect and learn. So, properties on circles, if you want to start numbering it and start learning then, can start doing this: if AB is the diameter, now this is anyway based on a triangle only. This is nothing but a right angle triangle. This will be right angle triangle when AB is the diameter. This will be 90. Now second one is here, where you consider four points on the circle, i.e. ADBC is a cyclic quadrilateral. Over there, sum of these two, sum of these two, d + c = 180, and this is called cyclic quadrilateral, because all four are on the circle. All the four points are on the circle. I am just taking you through the basic properties. Now the third one is, instead of diameter if you consider the secant over here, now, if you drop a perpendicular, i.e. again we are talking about right angle triangles here, both are right angle triangles here. So, if this is a perpendicular then it will get bisected at this particular point, i.e. AM will be equal to MB. You don’t need to write, you can visualize that anyway in the diagram. Now the fourth one can be almost similar. You take CD on the other side and drop a perpendicular. In that case, if, if OM equal to ON, i.e. if these two lines are equal, that implies AB = CD. This AB and CD they will be equidistant from the, if it’s equidistant then AB will be equal to CD. It’s fourth one. And these are all inter related. Now, next you drop this line further down, then this AB will become a tangent, so this is another right angle triangle. We are talking about lot of right angle triangles in circles. So, this is tangent perpendicularity theorem. This line is called a tangent because it’s just touching the circle at one point. This is the radius and together it will be right angle triangle. That’s five. Now sixth one is, from an external point if you drop two tangents they’ll be equal. That’s the next one. From an external tangent if you drop two points, they’ll be equal. Now these are, now how we’ll end up using this is basically you’ve a circle. These are all very basic properties but you can always make good questions out of this. You have a point outside the circle, you drop two tangents, combine with two radii.