Importance of Figures in Problem Solving vs Data Sufficiency | Geometry

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

(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. Answer (B).

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 $\sqrt64 = 8$ and area of triangle $QRT$ will be $\frac{1}{2} \times8 \times8 = 32$

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 $ABC = \frac{1}{2}\times AD \times BC$

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.

Geometry

Geometry

December 14, 2021

Geometry

March 25, 2023

Geometry – Importance of Figures in Problem Solving vs Data Sufficiency

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