Getting Geometry Problems on GMAT Data Sufficiency Wrong?
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I had a student recently whose Data Sufficiency (DS) accuracy was generally very high, and whose knowledge of geometry rules was solid. And yet, she was getting almost every geometry DS question wrong on practice tests!
This is actually very common: students who are otherwise good at geometry and/or DS struggle when the two things are put together.
Here are the 3 main reasons that students miss geometry DS problems:
- They trust their eyes too much
Consider this problem:
If we want to know the area of the circle, we need to know the radius. Since we have a square inscribed in the circle, the diameter of the circle would be the diagonal of the square:
Therefore, if we have any length of the square – side length or diagonal – we can find the diameter of the circle, and thus the area. Statement (1) gives us a side length, and statement (2) gives the diagonal, so they’re both sufficient. The answer is D. Right?
Wrong.
There’s a major assumption that we just fell for here: we don’t know for sure that it’s a square! It looks like one, and it probably is, but we can’t assume anything. Some of the sides might be imperceptibly shorter than the others.
Just knowing one side length, as in statement (1), would not be enough to find the diameter if ABCD is not a square.
Statement (2) tells us that the line from A to C is the hypotenuse of a right triangle. By definition, the hypotenuse of any right triangle inscribed in a circle is also the diameter of the circle. If we have that diameter, we can solve for the radius and thus the area of the circle. The correct answer is B.
You have to prove it.
Don’t just rely on your eyes! Ask yourself if they’re given you enough information to PROVE that the shape is what it appears to be.
- They don’t unpack the diagram before going to the statements
Try this problem:
You probably memorized the rule that the area of a rhombus is (diagonal1 × diagonal2)/2. So you might start plugging the statement information into the problem to see if you can get the length of BG and AC, CE and DF.
If that’s the case, you might end up with E, which is a wrong answer.
Instead, start by asking yourself “why did they give me all of this given information?” If all we care about is the area of each rhombus, why did they bother giving us an equilateral triangle?? Before you dive into the statements, make every inference you can:
- If CFG is equilateral, then CF = CG = GF.
- CG and CF are both sides of a rhombus, so every line segment up there is equal.
- All of the angles of CFG are 60°.
- That means angles AGC and CFE are 120°.
- Angles BAG, BCG, DCF, and DEF must all be 60°.
We could split each rhombus into 2 equilateral triangles:
- Each of those triangles would have the same area as CFG
- We can split an equilateral triangle into a 30-60-90:
- So, if we have a side length, we’ll know the height: (s/2)√3
- To find the area of a rhombus, all we’d need is the length of any side of any triangle, or the area of any triangle!
Statement (1) gives us the area of the equilateral triangle. The area of each rhombus will be double that. Sufficient!
Statement (2) gives us a side length of one of the triangles. As we inferred, that’s enough. Sufficient! The correct answer is D.
Rephrasing in DS geometry = unpacking the diagram.
Even though it’s natural to want to plug all of the given information into the problem on geometry, it’s dangerous to dive into the statements right away. There may be a great deal of information already inferable from the diagram and the given information. You may think you need information from the statements that you could already have inferred from the diagram!
Always yourself – why is this piece of given information here? The GMAT will never give you anything in the question stem that’s not necessary to the problem.
- They only draw one diagram
In other DS problems (algebra, number properties, etc), you know to test cases to see if a statement is sufficient. You’d test one number, see what you get, then test another number to see if you get the same result.
For some reason, people don’t apply this same strategy to geometry. They draw one figure, then just stick with it. Instead, you want to try what we’ll call the Rubber Band Geometry Technique: imagine stretching and pulling the figure in different directions, as if it were made out of a Rubber Band.
Try this problem from the Manhattan Prep Advanced Quant Guide:
If you simply draw the first trapezoid that comes to mind, you might think that you have sufficient information with either of the statements alone. Instead, you have to think about all of the different ways that a circle could be tangent to 3 sides of a symmetrical trapezoid:
Statement (1): if the circle is tangent to both parallel sides (Figure A or B), then the diameter would be 10. But if the shape corresponds to figure C or D, the diameter would be less than 10. Insufficient.
Statement (2): Knowing the length of the shorter side is not sufficient. In Figures A & B, the diameter of the circle is less than the length of the shortest side. In Figures C and D, it’s greater. Insufficient.
Both statements: If the height is 10 but the shortest parallel side is 15, then Figures C and D are impossible. We’re left with Figures A and B, each of whose diameter is the same as the height: 10. Sufficient.
This problem would likely be impossible to get right without drawing the array of all possible configurations.
If they don’t give you a figure, try to draw several different ones with the given information. If they give you a figure, don’t assume it’s exactly to scale. You can still draw stretched or squished versions. Trapezoids, rectangles, isosceles triangles, and many other shapes can come in different dimensions. ?
How to improve on DS geometry:
Avoid these 3 common pitfalls by doing the following:
- Never assume that the figure is what it appears to be, unless you’re given enough information to infer it. Be skeptical of what you see!
- Unpack the given information fully before diving into the statements. Ask yourself if you’ve made inferences from every single piece of given information.
- Draw different versions of the figure, if possible. Ask: can I stretch or squish it?
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Céilidh Erickson is a Manhattan Prep instructor based on New York City. When she tells people that her name is pronounced “kay-lee,” she often gets puzzled looks. Céilidh is a graduate of Princeton University, where she majored in comparative literature. After graduation, tutoring was always the job that bought her the greatest joy and challenge, so she decided to make it her full-time job. Check out Céilidh’s upcoming GMAT courses (she scored a 760, so you’re in great hands).