Concerning the point $(7,a)$ on the line containing $(0,0)$ and $(4,2)$

Question

I have recently been studying to take the GRE's and while working through the math section I find a lot of problems similar to this:

Now I know it is supposed to be assumed that point $O$ is marked at the origin and thus (0,0) is a point on the line and you can then conclude that the answer is B. But no where in the problem does it state that that point $O$ is the origin or even that points (4,2) and (7,a) fall on the same line and as such I feel that one could make a very convincing arguement that D is a correct choice as well. This is just one instance where you are expected to assume a good amount of information based on a picture

So my question is am I right to be critical of the way these GRE questions involving pictures are presented or am I just being far to critical of these questions.

Answer 1

Mark Fischler is incorrect - the GRE is not about figuring out "what the examiner intends". There are certain conventions that the GRE follows, and certain instructions given before each section, that dictate how the problems work. Here are the instructions that come at the beginning of every Quantitative section:

"All figures are assumed to lie in a plane unless otherwise indicated.

Geometric figures, such as lines, circles, triangles, and quadrilaterals, are not necessarily drawn to scale. That is, you should not assume that quantities such as lengths and angle measures are as they appear in a figure. You should assume, however, that lines shown as straight are actually straight, points on a line are in the order shown, and more generally, all geometric objects are in the relative positions shown. For questions with geometric figures, you should base your answers on geometric reasoning, not on estimating or comparing quantities from how they are drawn in the geometric figure.

Coordinate systems, such as x y planes and number lines, are drawn to scale; therefore, you can read, estimate, or compare quantities in such figures from how they are drawn in the coordinate system.

Graphical data presentations, such as bar graphs, circle graphs, and line graphs, are drawn to scale; therefore, you can read, estimate, or compare data values from how they are drawn in the graphical data presentation."

In addition, there are more conventions and assumptions that the GRE follows that can be found here: https://www.ets.org/s/gre/pdf/gre_math_conventions.pdf

This includes that for an x-y plane the axes intersect at the origin. $O$ can also be used to denote the center of a circle, but if it's for an x-y plane it's the origin.

It's important to note that the ETS (who writes the exams), writes questions based off of these math conventions such that there is only one correct answer. If you're looking at a question and thinking, "But what if this wasn't like that..." then you're likely contradicting one of the conventions in the pdf above.

Source: ETS website, personal experience (been teaching GRE for 5+ years)

Answer 2

Part of the question is whether you can figure out what the examiner intends, based on the (possibly mildly incomplete) information given. In the problem shown, any reasonable person would assume the dots are intended to dis-ambiguate the meanings of the otherwise ad hoc (4,2) and (7,a), and that the axes given are intended to represent the fact that the line passes through the origin.

Thus if you give answer (D), you may feel self-righteous but as a practical matter you will be graded as incorrect, and with considerable (though not absolute) justification.

Answer 3

If someone were to simply post the graph without context on this site and asked to determine (if possible) whether $a < 5, a = 5,$ or $a > 5,$ people might justifiably ask for clarification of the same points you find dubious in this problem.

In context, however, this is a GRE problem. As you observed, there are a lot of problems similar to this one, using a similar graphical shorthand for the problem conditions. In that context, you can (and should) assume that new problems will use the same graphical shorthand.

For example, with regard to the line passing through the point $(0,0),$ it's been long enough since I took the GREs that I don't recall whether $O$ is consistently used for the origin or is sometimes used for the center of a circle. But you also have lines with arrowheads and the letters $x$ and $y,$ which are common graphical shorthand for saying, "These are the $x$- and $y$-axes." The point where they intersect is the origin.

The GREs are not a pure test of general ability; strictly speaking, they're a test of the ability to take GREs, which people hope is at least somewhat correlated with other abilities. That is why it is wise to study for them; there is a specialized skill you have to develop and retain for at least long enough to complete the exam.