Using the information given in diagrams to your advantage
The following sentences appear in the directions to the GMAT Problem Solving questions.
A figure accompanying a problem solving questions is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted.
Many GMAT-takers underestimate the valuable information given there. Diagrams on GMAT Problem Solving are basically drawn to scale. The only time that doesn’t hold is if you see the note printed “Diagram not necessarily to scale” — then, all bets are off about how the figure actually looks. But if that disclaimer is not printed, what you have on GMAT Problem-Solving is a diagram drawn to scale, guaranteed.
What You Can Assume
Consider the following question:
1) The area of rectangle ABCD is closest to which of the following?
Suppose we don’t know the math to answer this question. We are told it’s a rectangle, so we know the angles must be right angles, and we know the area must be length (AD) times height (AB). We know the height is 10. We know AD is drawn to scale. It definitely is longer than AB, so the area is definitely larger than 10 x 10 (answer (A) is out). AD doesn’t look as long as twice AB, so the area is definitely less than 10 x 20 (answers (D) & (E) are out). Notice, with pure spatial estimation, we eliminated three of the five answer choices, so it will be to our advantage to guess randomly from the remaining two if we can’t decide between them. Estimating from size can be a huge help if you don’t remember the way to solve the problem.
BTW, the real math solution to that question: from the properties of the 30-60-90 triangle (ACD), we know that AD = 10*sqrt(3), and since sqrt(3) is approximately 1.7, AD is approximately 17, and the area is approximately 170. Answer = C.
We can assume that a line in a diagram is straight — if the line goes through points A & B & C, and it looks like those three points are “collinear,” then they indeed are, and the line is straight. We have to be careful here: some students mistakenly us the word “straight line” as a synonym for “horizontal line”: this is a grave misunderstanding, and leads to significant confusion. We absolutely cannot assume that lines are perfectly horizontal or vertical from a diagram, but we absolutely can assume that they are straight, i.e. collinear.
Here’s another. This is from the GMAT OG. In the GMAT OG12e, it’s Problem Solving #210, and in the OG13e, it’s Problem Solving #211.
2) In the coordinate system above, which of the following is the equation of line l?
(A) 2x – 3y = 6
(B) 2x + 3y = 6
(C) 3x + 2y = 6
(D) 2x – 3y = –6
(E) 3x – 2y =–6
A student asked about this question: how do we know that the x-intercept of line l is 3 and the y-intercept is 2? Well, technically, we don’t know that they are exactly 3 and 2, but we know from the diagram that if they are not exactly 3 and 2, they are very very close. Thus, x-intercept = 3 and y-intercept= 2 make an excellent starting point: even if they are not spot-on correct, they are very good approximations. As it happens, the exact values themselves lead to the correct answer of B.
What You Can’t Assume
You can’t assume lines are parallel, because many special properties are true only if two lines are exactly parallel, but the naked eye cannot distinguish exactly parallel from almost parallel. For example:
These lines look parallel, right? They’re not: they are 1/10 of one degree off from exactly parallel, and that means: none of the special geometry facts for parallel lines would apply to these lines.
The same applies to right angles. An angle of 89.9º or 90.1º will look like a right angle to the unaided eye, but if it’s not an exact right angle, none of the special right angle facts (like the Pythagorean Theorem) will apply. For example:
These are two squares, right? Think again. Here is each one with individual measurements:
ABCD is actually a rhombus: four equal sides, and opposite pairs of angles equal, but not equiangular, the way a square should be.
EFGH is actually an isosceles trapezoid: equal pairs of base angles, and the legs (EF & GH) are congruent. Both look like squares, but neither one is.
None of the parallel properties in geometry are true for “almost parallel,” and none of the right angle properties are true for “almost a right angle.”
Diagrams on GMAT Problem Solving are drawn to scale. That serves you very well when you are approximating. That doesn’t help you, and may mislead you, if you need something to be exactly true.
Here’s a practice PS question with a diagram drawn to scale: