GRE Geometry: The Impossible Task
In one of my recent classes, I told the students “You’ll never know how to answer a geometry question.” The reaction was fairly predictable: “Why would you say that?!? That’s so discouraging!!”
Of course, I certainly was NOT trying to discourage them. I used that statement to illustrate that geometry questions are often a type of quantitative question that can feel immensely frustrating! You know what shape you have, you know what quantity the question wants, but you have no idea how to solve for that quantity.
This is what I meant when I said you’ll never know how to answer these questions. That “leap” to the correct answer is impossible. You can’t get to the answer in one step, but that’s all right: you’re not supposed to!
(An important aside: if you’ve read my post regarding calculation v. principle on the GRE, you should be aware that I am discussing the calculation heavy geometry questions in this post.)
The efficient, effective approach to a calculation-based geometry question is NOT to try and jump to the final answer, but instead to simply move to the next “piece”. For example, let’s say a geometry question gives me an isosceles triangle with two angles equaling x. I don’t know what x is, and I don’t know how to use it to find the answer to the question. But I DO know that the third angle is 180-2x.
That’s the game. Find the next little piece. And the piece after that. And the piece after that. Let’s see an example.
The correct response to this problem is “Bu-whah??? I know nothing about the large circle!”
But you do know the area of the smaller circle. What piece will that give you? Ok, you say, area gives me the radius. A = pi*r^2, so pi = pi*r^2, so r^2 = 1, so r = 1. Done, and let’s put that in the diagram.
(By the way, did you build your own diagram on your paper? I suppose you could try to do this all in your head, but… why do you hate yourself?)
So now we have the radius, so that gives the diameter. D = 2. Let’s put that in the diagram.
Now what? Well, looks like that diameter is also the length of a side of that square. What else can you draw? What if you draw the diagonal of the square, and solve for that? Now you can either use the Pythagorean theorem, or you can just know that the sides of a 45 : 45 : 90 triangle always follow the proportion x : x : ?2. And you can put that in your diagram.
At which point, you have the diameter of the larger circle, which will give you the radius, which will give you the area.
But how do I get fast at this? That took forever!
Good question! You get faster with practice: practice beyond the bounds of the question. Can you find the area of the square? What about the difference between the area of the square and the area of the larger circle? What about the circumference of the circles? Or the perimeter and area of the right triangle formed by the diameter of the square?
Whenever you review a geometry problem, try to do more! More than what the question asked, with more shapes.
Go beyond the question. What if you drew a line segment from Q to R? What kind of triangles will you create? What is the measure of the central angles? The measure of angle PQO? Can you solve for the length of PQ?
This, my GRE compatriots, is how you get fast.
