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Geometry - Importance of Figures in Problem Solving vs Data Sufficiency

Many test takers, though good at Math find Data Sufficiency difficult. They are much more used to the straight forward Problem Solving pattern. The very principles behind the two question types are very different.


In Problem Solving questions, our target is to find just one solution. For example, when we have questions involving percentages, we assume some values and get the answer. No matter what values we assume, we will always get the same answer as long as the integrity of the data is maintained.


In Data Sufficiency questions, our target is to find multiple possible solutions  after using all the given data and arrive at answer (E). If we are unable to find more than 1 solution using either statement (1) and/or statement (2), we arrive at answers (A), (B), (C) or (D).


The aim is diametrically opposite in the two cases. Therefore, our strategies in the two cases would also be different and they are. Consider Geometry questions with figures in them. In Problem Solving questions, we try to make the figures as symmetrical as possible under the given constraints. With symmetrical figures, it is easier to get an answer. One answer is all we need.


In Data Sufficiency questions, we try to make the figures as extreme as possible. Only the given data should hold in such a figure and no symmetry should exist in the other dimensions. Only then will we be able to really figure out whether the given information is enough to arrive at a unique answer.

Let’s explain this using two examples:


Problem Solving Question


In the figure above, the area of square PQRS is 64. What is the area of triangle QRT?

(A) 48
(B) 32
(C) 24
(D) 16
(E) 8


This is a Problem Solving question.


All we are given is that PQRS is a square. Note that the location of point T is not defined. It is just any point on side PS. We can place it anywhere we like as long as it is on PS. At what point will it be easy for us to calculate the area of triangle QRT? Of course, T could be the middle point of PS (bringing in symmetry) and we could calculate the area of the triangle or we could make it coincide with S so that QRT is a right triangle half of square PQRS. Then, the area of triangle QRT will simply be half of 64, i.e. 32.


Note that we don’t necessarily need to do this. We can assume T to be a random point, drop an altitude from T to QR, find that the length of the altitude will be same as the side of the square, find that side of the square will be √(64) = 8 and area of triangle QRT will be (1/2)*8*8 = 32


We will arrive at the same answer of course! But, assuming a better position for point T (but only because it is not defined) will cut the calculations and help us arrive directly at 32 from 64.


Data Sufficiency Question



If AD is 6 and ADC is a right angle, what is the area of triangular region ABC?

Statement 1: Angle ABD = 60°
Statement 2: AC = 12


Looking at the figure, many test takers are tempted to think that the altitude AD will bisect BC. Note that that may not be the case.


According to the data given in the question stem alone, the figure could very well look something like this:



All we know is that ADC is a right angle and the length of the altitude is 6. We don’t know whether any of the sides are equal, etc. Hence, it is a good idea to redraw the figure with extreme proportions – one side much greater than the other.


Now we can use the given statements to re-adjust the proportions.

Area of triangle ABC = (1/2)*AD*BC

We know that AD is 6. But we don’t know BC. Let’s examine each of the statements separately.


Statement 1: Angle ABD = 60°

This statement tells us that triangle ABD is a 30-60-90 triangle. Knowing the length of AD will give us the length of the other two sides too. But here is the problem – to know BC, we need to know length of CD too. That we cannot find from this statement alone. This statement alone is not sufficient to answer the question.


Statement 2: AC = 12

We know that ADC is a right angled triangle. Knowing AC and AD, we can find the length of CD using Pythagorean Theorem. But we cannot find BD using this statement and that is needed to get the length of BC. This statement alone is also not sufficient to answer the question.


Using both statements, we can find the lengths of both BD and CD, and hence, can find the length of BC. This will give us the area of the triangle. Therefore, our answer is C.


Note here that if we mistakenly assume that D is the mid point of BC, we might come to the conclusion that each statement alone is sufficient and might mark the answer as D, instead of C. Hence, it is a good idea to redraw the given figure in a Data Sufficiency question to ensure that it has as little symmetry as possible.


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