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Geometry - Making Assumptions on Problem Solving

Today we will discuss the flip side of “do not assume anything in Data Sufficiency” i.e. we will discuss “go ahead and assume in Problem Solving!”


Problem solving questions have five definite options, that is, “cannot be determined” and “data not sufficient” are not given as options. So this means that in all cases, data is sufficient for us to answer the question. So as long as the data we assume conforms to all the data given in the question, we are free to assume and make the problem simpler for ourselves. The concept is not new – you have been already doing it all along – every time you assume the total to be 100 in percentage questions or the value of n to be 0 or 1, you are assuming that as long as your assumed data conforms to the data given, the relation should hold for every value of the unknown. So the relation should be the same when n is 0 and also the same when n is 1.


Now all you have to do is go a step further and, using the same concept, assume that the given figure is more symmetrical than may seem. The reason is that say, you want to find the value of x. Since in problem solving questions, you are required to find a single unique value of x, the value will stay the same even if you make the figure more symmetrical – provided it conforms to the given data.


Let us give an example from Official Guide 13th edition to show you what we mean:


Question: In the figure shown, what is the value of v+x+y+z+w?


(A) 45

(B) 90

(C) 180

(D) 270

(E) 360


We see that the leg with the angle w seems a bit narrower – i.e. the star does not look symmetrical. But the good news is that we can assume it to be symmetrical because we are not given that angle w is smaller than the other angles.  We can do this because the value of v+x+y+z+w would be unique. So whether w is much smaller than the other angles or almost the same, it doesn’t matter to us. The total sum will remain the same. Whatever is the total sum when w is very close to the other angles, will also be the sum when w is much smaller. So for our convenience, we can assume that all the angles are the same.


Now it is very simple to solve. Imagine that the star is inscribed in a circle.

Star in a circle


Now, arc MN subtends the angle w at the circumference of the circle; this angle w will be half of the central angle subtended by MN (by the central angle theorem discussed in your book).


Arc NP subtends angle v at the circumference of the circle; this angle v will be half of the central angle subtended by NP and so on for all the arcs which form the full circle i.e. PQ, QR and RM.


All the central angles combined measure 360 degrees so all the subtended angles w + v + x + y + z will add up to half of it i.e. 360/2 = 180.


Answer (C)


There are many other ways of solving this question including long winded algebraic methods but this is the best method, in my opinion.


This was possible because we assumed that the figure is symmetrical, which we can in problem solving questions!


But beware of question prompts which look like this:

– Which of the following cannot be the value of x?

– Which of the following must be true?


You cannot assume anything here since we are not looking for a unique value that exists. If a bunch of values are possible for x, then x will take different values in different circumstances.


If we know that the unknown has a unique value, then we are free to assume as long as we are working under the constraints of the question. Finally, we would like to mention here that this is a relatively advanced technique. Use it only if you understand fully when and what you can assume.


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