CAT Coaching – Understanding Geometry – Part4
Okay, sixth one, if you see, you can see 4 points on the circle and so we know that even the lines are joined here. One possibility is that it’s based on cyclic quadrilateral, and basic data given is this is 30, it’s given that these two lines are parallel, plus these two sides are equal. Now when you have two sides equal, opposite angles will be equal, so this will be 30. Since these two lines are parallel, what will be this? Again be 30, so this is 60, so the required angle is 120. It’s and easy one, right? Based on, why is it 120? Because it’s a cyclic quadrilateral. Seven one is a, no, seventh one is an area based question, where, the ratio is given in such a way that BC is 2 times AD, right? In question number 7, BC is 2 times AD, overall area is given as 54. BC is 2 times AD is nothing but you’ve x here and 2x here. If you just double the diagram it becomes easy for you to visualize. Just double of this diagram is a hexagon. Right? This is just half of a regular hexagon. That’s where you’ll find this 1 is to 2 ratio. So, this ABCD is a trapezium and double of that is actually a regular hexagon. Now it’s easy because what is given is, this is a midpoint, this is another midpoint. So, just cut the diagram, and you can easily find the answer. If I divide the diagram like this, this is a parallelogram, this will be a parallelogram. And if I join the diagonals, these two will be equal. If I take it the other way around, this is also a parallelogram, and I join the midpoints here these two will be equal. So effectively all these three are equal. And since you’re joining the midpoints, again it will become equal. If you’re joining the midpoints median anyway will divide into equal areas again. So, all of them are equal. So 6 equal areas. Total is 54. Question is to find the difference between this and this. Which is, here it’s 2 right? And here it’s 1. 54, if you divide it into 6 equal areas, its 9. Question is to find the difference between 18 and 9. So should find the difference between 18 and 9. So, the answer is option a), question number 7. And you need not, and it’s a, this is an area based question. Sixth one is an angle based question. First classification is like that only. You check whether it’s an angle based, area based or length based question. Then, you can start solving them faster. So, the answer is option a). Double of this diagram is nothing but, you get something like this. This is just double the diagram, and after that this here, whatever you divide, whether you divide it like this, these are all 12 equal areas. Now, you can divide like this also, 12 equal areas. They will all be equal.
Next question. Now see. Just one more thing. When you have a circle, and right angle triangle with value is given, 45-45-90 or 30-60-90. This is important. Now, 45-45-90 triangle, what is the side ratio? 1:√3:2. And in a 30, sorry! 1:1:√2. And the 30-60-90 triangle, sides are in the ratio 1:√3:2. Now, the best part is knowing this, is one thing, but knowing that this is the, this is the only two ways you can get heights and distances questions or angle to side or side to angle ratio questions. That will make a big difference when you solve questions. You’ll get either this or this. Because anyway don’t expect you to, why not 15-75-90 because they don’t expect you to memorize values of this. Neither they will let you use an algorithmic table or a scientific calculator. So that is, so that is why you can actually expect, if not the normal pythagonal replied questions. If it’s a side to angle, angle to side ratio questions, this is based on basic trigonometry you’ll get either based on 30-60-90 or 45-45-90. And only this. Knowing that only this is what you’ll get is a, will make a big difference when you solve. Because you’ll look forward to this and not the other way around. I’ll show you so many questions where you keep these two things in mind, you’ll solve it much faster. You can almost predict that it’s either 30-60-90 or 45-45-90. In what kind of questions? Angle to side, or side to angle questions. Now this question is nothing but, you’re supposed to find out, angles are given in a right angle triangle. This question is about finding some sides. It can be the other way around also. And typical pattern questions you can expect. Side to angle, angle to side questions in right angle triangles and there are lot of them in you’ll come across in circles, right? There are so many ways you can get a right angle triangle in a circle, so many ways. Like this is a right angle triangle. Like this is a right angle triangle. Like this, this is a right angle triangle. So many, where you get right angle triangles. Then, lines and circles intersect in one way or the other. So, and in these kind of questions, side to angle questions, or vice versa, in a right angle triangle these are things which you can directly expect and mark. So the moment you see 45-45-90 here, you know that it’s based on, this is the question right, you have 45, 45, 45, 90. Now, see the points which are defined, right. That’s very important. You see the point, concentrate on the points defined. 1, 2, 3, 4. So it’s not that you’ll miss out. You’ll immediately take, you’ll think about cyclic and you’ll take this as 90. So it’s not that these are the same things which you’ve learnt before but how you think is everything when you see the diagram, so you’ll solve it. One is you can directly take this 90, because see the diagram where points are defined, 4 on the circle. There are not many ways which you need to think, right, 4 on the circle, 3 on the circle 1 inside, 3 on circle 1 outside, 2 on circle 1 inside, 1 outside. All the important properties which you can use and get the next one, is covered. So, it’s easy but it’s easier than you think. Though you have 90 and 90, then in, now it’s over because this is 90, this is 45, this is 45. This is given as 5 which is opposite 45. So, this is 5 right? This is given as root, 6 √2. Now see this triangle, the inner one, this is a 45-45-90 triangle. So opposite 45 its 5, so opposite 90 will be 5√2, and opposite 40 this will be 5. So this distance is 11√2. EC is 11√2. Which is, now look at the outer triangle. That is again a 45-45-90 triangle. So opposite 45 its 11√2. Now this is a very simple question right. Just angle to side question. This will be 11√2. And opposite 90 it will become √2 into 11√2 which is 22. So finding this distance, as 17, is an easy one. And both, both 6 and 8, can see it in previous papers. So this is a normal level of questions you will expect angle based, angle to side, side to angle. But there is nothing which you can’t expect here. It’s a 40-40, 45-45-90 question. You can get 30-60 questions based on 30-60-90 also. Okay, so answer here is, what is the answer, option? Question is not to find the complete distance right? It’s only to find, its 17 right? Answer is 17. Some of the sides are given. Something like this, you have, this is 45, sorry, 90. Opposite 90 you’ll get 5√2. That’s the, this is the only thing which you need to find right? For that you’re taking this as 90. Because 4 points are on the circle. Four are on the circle, so immediately you think about cyclic. 4 points on the circle, it’s not that there are, you’ll immediately think about. This is everything. You see 4 points, you either take like this, or you take it like this, in terms of angle, or even in terms of this right? This into this is this into this, because these are similar.
Okay, here, first question you can assume any equilateral, any right angle triangle, because that’s based on right, any right angle triangle, right, and how, the best assumption will be like you can take 3:4:5 or 6:8:10. And when I write 6:8:10, see this is the pattern which I have told you right, how the, how all the values are related in a 3:4:5 triangle. If it is just 3:4:5, i.e. just a multiple of 1 here, then r = 1 itself, and R = 2.5. Now, if it is 6:8:10, if the assumption is 6:8:10 then r will be 2, because this is actually a multiple of 2, and R will be 2.5 times which is 5. So using this you can actually eliminate all the wrong options. Using this you can eliminate wrong options and that’s the only way you should solve. You, unless and until you know the direct property and you’ve memorized it where you can mark the answer, because all this, anything which you’re writing in variables is because it’s a property. Maybe not for ever triangle, but for that perpendicular set of triangles, you can assume any triangle which satisfies the definition. Here it’s a right angle triangle, so you can take 6:8:10. When you’re taking 6:8:10, you’re effectively taking r = 2, and circumradius as 5. So this is what you need to substitute. And eliminate where its, wherever it’s not matching. And circumradius why it’s 5 and all is, you can visualize right? Initially we discussed in our right angle triangle it will be 5 and 5. It will be half of the hypotenuse. And inradius is nothing but the same ratio with which you’re multiplying this 3:4:5. So, that is why assumption won’t take any time, so it’s better to know some few basic values so you’ll write r = 2, and R = 5, in a triangle like this where 6:8:10. So what is answer? 2 times, now the triangle is defined right? You’ll have to take in the same order. C is at 90. This you should keep it correctly. C is 90, so, you can take ABC. So, opposite this, this is small a, side a, this is side b, b = 6, a = 8. So, you can directly mark the answer where it’s matching. Where it’s not matching, that’s 2 times, 2 times 7 is 14. 14 is 6 + 8. Other options are not matching. So, the idea here is, this is, this is a flexible question. Just like question number 1, just like question number 3. And, 14 is, that’s a direct question right? 14. What’s the answer? 14. Only thing is. Okay, this is given. 20 and 40. Something which you’ll get direct out of this is 20 + 40 is this, right? 20, sum of, this is the same thing which we use for proving that basic property right, where 1 + 1 is 2. Sum of remote interior angles is exterior angles. Very, very basic of triangles right. So, 20 + 40 is 60, then it’s over. After that you can just see now the question is based on something inside the triangle right? You need to find out some angle inside, so, 4 points are there. 4 points I told you it can be about, so this is 60 means, this is 60, so this will be 60. This is what you need to find out. Remaining is 30. Out of 90 right because when this, this is the center right? So when diameter is involved, we know that its 90. So, whenever it’s a diameter you should immediately think about a right angle triangle out of which part of it is given as 60, remaining is 30. Remaining is 30. And you’ll anyway think about this when you see 4 points, you’ll anyway think about either this, or the inscribed angle is equal in the same arc. Unless and until, if it’s a length based question you should immediately think about this into this as this into this. These are all angle based questions.
Now, see question number 17, 18, 22, I’ve already discussed 22. And some of the questions in between questions are also already covered. 17, 18, 22, 23, 25. I’ve already explained one or two here. You can just mark the answers.
Okay, 22 what’s the answer? 22? Something which I have already explained right? This is the same as, you start with a circle, then take, I’m just taking some, just 4 tangents all at right angles. So the next circle will, this is how you’ll visualize that what is given in the question. This is given as r, so this is r, so what is this distance? So what’s your answer for question number 22? Option a). Question number 22 is option a). This is √2r and this is 2 times right? Maximum distance of MN is nothing but, MN is nothing but, it’s the next diameter, i.e. the next circle and the diameter. And in this perpendicular series of square circle square circle and you’ll find good number of questions based on this perpendicular diagram. And repeating this. Because in a way they are all geometric progressions. Area-wise, perimeter-wise, they are all, they will all fall into geometric progressions. Because this is, these are all regular diagrams. Since they are all regular diagrams you’ll keep on multiplying by the same number only. Whether you look at area or perimeter. So, 22 is option a).
Now, 23 is right angle triangle question given in words. Something like this, something like this, this is given as 30. Here the speed is given as 2. And these are all predictable right? Everything you can predict. This is a heights and distances question. 90 here, so this will be 1. The question is as easy as, there is an angle to, all heights and distances questions are angle to side conversions only. All heights and distances questions. They are not worth preparing for because of that. They are all side, angle to side conversion questions. That too, even before you start the question you know that it’s either 30-60-90, or 45-45-90. Sometimes in, supposedly difficult questions, both. Both will be together. Can put both the triangles together right? You can have both the triangles together. One is 45-45-90, the other one is 30-60-90. So, here it’s like this: opposite 30, this is given as 30. Just that it’s given in words. So along this line the speed is 2. So along this line speed will be 1. This is 90, and this is the wall. So, 30-60-90 triangle. Sides are in the ratio 1: √3:2.
Now, they are all direct questions, which, we’ll find heights and distances questions again. All of them are round 1 questions and sure shot answers. There is no way you’ll miss out. You’ll have to struggle really hard to make a mistake. So, 25, already discussed, right? We have already seen that question. That’s a, that’s a direct question. And what is important is, this is a length based question, and we know that, that is given here is circle and 4 points outside. So you’ve 6 here, 7 here, 4 here, 3 here. Now, next while discussing 17 together is, this is also a similar question. Because it is again length based. Length based where the 3 points defined are outside the circle. See, angle based means what we’ll do is, here also you can make angle based questions. You’ll immediately start doing this, start doing this, and over. 3 kites. Length based you’ll anyway look at the length based property right. What is the property? Tangents are of equal length. So there is no, you’re not leaving anything to chance. These are all, almost something which you can directly expect. So, this is given as x, so this will be x. right? This will also be x. this distance. Now this will be 10 – x. by the time you start like this and by the time you finish one round, you’ll get the answer. So, 10 – x, this will be 10 – x, so that’s equal, and this will be 7 – 10 – x, which is x – 3. 7 minus of 10 minus x, so that’s x – 3. And this will be x – 3. Now it’s over right. You’ve x, x – 3 equal to 9. x + x – 3 = 9. Both 17 and 25 are actual questions. And good questions. But you know that it’s, they are all actually easy. So what’s the answer? x + x – 3 is 9, so you can directly mark the answer. x + x – 3 = 9, so x = 6. 17, 25 are similar questions. Only difference is one is circle and 3 points, other one is circle and 4 points. At this level you should start reading the question. That’s when it’s much easier than what much easier than otherwise. There are anyway easy ones. I’m very sure, lot of these questions you’ll anyway solve with or without preparation. Now you’ll solve it without any preparation. We’re not preparing. We are not solving hundreds of questions here. And I don’t want you to solve. You, you simply ask so many students are giving feedback right. 9 out of 10 who’ve got into IMs after attending, sitting in the auditorium, they’ll tell that they didn’t prepare at all. The other 1 or 2, because some of us will only get the confidence only when we because we are used to practicing hours and hours and hours 100 of hours for all competitive exams. But this is not an exam. This is a test of common sense. So, at 10th standard level. Concepts used are at 10th standard level. That time maybe some of us thought these were difficult. Not anymore.
Now, next question is, and soon because if I just write polygon in a polygon, almost 50% of your questions will come under this category. Polygon in a polygon. Area based, length based, angle based. That’s what you are learning in plain geometry. This is a very basic polygon in a polygon question. Next one. Right? Now, so you might think solving it like this, i.e. taking 1, 1, 1, 1. This is a typical CAT question. An easy one. Anyone will solve it. So it’s all about can you do it faster. That’s the only question. Anyone will solve it. 8th standard students, 7th standard students, they’ll solve it. Area of the square minus are of the circle by 4, you’ll get region 1. And what is region 2? Area of the circle minus area of the square by 4. You’re dividing by 4 because there are 4 of them like that and you need to count only 1. Only time consuming part is π is involved in calculation. Options are without it. These many things you should immediately understand, π is not involved, options are without it, so it’ll take 1.5, 2, 2.5 minutes depending on your calculation speed. So, you’ll anyway get the answer. You’ll get this answer. You’ll get it much faster if you’re using flexibility. Now what is a flexibility? Its, it is flexible because these are all standard diagrams, these are all standard diagrams. So this is the original diagram, what you have on the left hand side. This is region 1, this is region 2. Now instead of doing that, this is the whole discussion, the whole discussion was about this only, where whether I take a square like this, or like this, nothing is going to change. Area is not changing anyway. You’re just turning it around little bit. You know what degree it is getting turned. But, whether you take it like this or like this, all these squares will be, these are all of the same ratio right. This ratio is never going to change. Circle to circle ratio is 2 by 1, circle to circle ratio is 2 by 1. Square to square ratio is 2 by 1. Square to circle ratio is 4 by π. These things will never change. Even if you change the orientation. So, that is why now the question is even much faster if you just shift it like this. Take the inner square like this. But for some reason, none of us think like this. And, this is region 1, this is region 2. Then the question is over. Because circle is out. It’s still the same question right? Circle is out. Now the answer is just 2 lines away. And we call it cut and count. You can call it whatever you want. But these are not questions at our level. Please don’t waste your time solving these kind of questions after 1 or 2 sessions. So, you’ll count the answer. Overall area is 256, and you’ll count the answer: 1, 1, 1, 1, all of them are equal; it’s 8 and 1 out of 8. You’ll get the same answer. So you might think this is the fastest method. You’ll soon realize you will do, you can do it even faster if you solve it through numbers, which I might have discussed. Learning diagrams through numbers is very much possible. So, this is not the fastest method. This is already faster than the guy who is making your paper. 100% again. You try giving this question to anyone and see how they are solving it. In a way the other way around. The way I’ve discussed before. So, and, this is not teaching the, I am not teaching anything, I am just making you think in multiple directions. Something which you have never done before. I didn’t teach you anything because it’s all, there is nothing to teach in, what will I, what will I teach in 8th standard, 9th standard, 10th standard geometry? So, to a lot of people who are used to clearing engineering math exams while studying for one day? I can’t teach anything at this level. I am just showing you how to learn and how to solve. How to think. Which I am very you’ll soon realize. So I didn’t teach you anything. That is there is nothing to go back and practice. In today’s session I can tell that with full confidence. Please don’t practice when it comes to geometry. There are areas where you need to actually go through when it comes to numbers.