CAT Coaching – Geometry
I’m going to show you how you can actually improve the way you think, when you look at diagrams… And you’ll understand that by the time I finish discussing this question.
So, let me give you the question… Take a square, with side 16… put the maximum circle inside… and inside that circle put the maximum square. Now, in this diagram, two regions are shaded… Let me number it as 1 and 2. You have to find Region 1 + Region 2.
I will also give you answer options… which are, 16, 32, 48, 64 … and if I give you time… I’m very sure… given time, all of you will solve this question… and there is a very good chance that 99% of you are solving it this way… and, how will you solve?
Region 1… How will you find Region 1 ..? Region 1 is, Area of the square – Area of this circle… and you’ll divide it by 4 …. Now, why you are dividing by 4 ? As you can see… there are four equal regions…
Same way… How will you find Region 2 ..? Region 2 is, Area of the circle – Area of the inner square… and there again you’ll divide by 4, because as you can see there are four regions like 2.
Now when you are doing this, Pi is involved in calculation, because circle is involved… Options are without Pi… that’s why it’ll end up taking 1 & 1/2, 2, 2 & 1/2, 3 minutes… depending on your calculation speed… whether you are fast / super fast / average… that will decide how much time you’ll take to convert this into the answer options.
So I’m sure you’ll get the answer.. This is the original question… If I’m asking question back, take a circle… whether I take a square like this… or whether I take a square like this… can I take any square..? can I take any square here..? I’ve just changed the orientation… I’m simply rotating the first one, square 2, get the second one… Now, if I use the same logic or same common sense to solve this question, what we’ll get is…
So let’s look at the original diagram… this is the diagram… the regions are shaded 1 and 2… Now the inner square… if I just rotate like this… what’s happening..? you get Region 1 here… you get Region 2 here… now they are together… So, now, it’s an advantage, because… circle is out… so Pi is out of calculation… Now it’s not a question..! It’s… It’ll become well below our level… at its lowest level you want to solve, how you are supposed to solve..? I’ll write in two words… (1) cut the diagram, and (2) count the answer… Cut the diagram like this… and count the answer 1, 2, 3, 4, 5, 6, 7, 8 … all of which only one region is shaded. So the answer is 1 by 8 of the outer area, anyway is 162 … So 1 by 8 of 256 …. you’ll get the answer as 32.
So this is exactly what I meant by… that even though all of us are capable of doing this… since most of us are tuned not to think… most of us would have solved it using the first method, which would have taken much more time… This is much better and faster… and this is how you are supposed to solve questions… especially when it comes to competitive exams where time is going to play a crucial role in deciding your rank and percentage… And I’m very sure that now all of you are thinking why I didn’t do it this way before… Because it’s… it’s that easy and, this is just an example where…of… how, by thinking better you can actually simplify… whether it’s questions or concepts.