3 Steps to Better Geometry
A couple of months ago, we talked about what to do when a geometry problem pops up on the screen.
Do you remember the basic steps? Try to implement them on the below GMATPrep problem from the free tests.
* In the xy-plane, what is the y-intercept of line L?(1) The slope of line L is 3 times its y-intercept
(2) The x-intercept of line L is [pmath]-1/3[/pmath]
My title (3 Steps to Better Geometry) is doing double-duty. First, heres the general 3-step process for any quant problem, geometry included:
All geometry problems also have three standard strategies that fit into that process.
First, pick up your pen and start drawing! If they give you a diagram, redraw it on your scrap paper. If they dont (as in the above problem), draw yourself a diagram anyway. This is part of your Glance-Read-Jot step.
Second, identify the wanted element and mark this element on your diagram. Youll do this as part of the Glance-Read-Jot step, but do it last so that it leads you into the Reflect-Organize stage. Where am I trying to go? How can I get there?
Third, start Working! Infer from the given information. Geometry on the GMAT can be a bit like the proofs that we learned to do in high school. Youre given a couple of pieces of info to start and you have to figure out the 4 or 5 steps that will get you over to the answer, or what youre trying to prove.
Lets dive into this problem. Theyre talking about a coordinate plane, so you know the first step: draw a coordinate plane on your scrap paper. The question indicates that theres a line L, but you dont know anything else about it, so you cant actually draw it. You do know, though, that they want to know the y-intercept. What does that mean?
They want to know where line L crosses the y-axis. What are the possibilities?
Infinite, really. The line could slant up or down or it could be horizontal. In any of those cases, it could cross anywhere. In fact, the line could even be vertical, in which case it would either be right on the y-axis or it wouldnt cross the y-axis at all. Hmm.
Make some kind of symbol on your diagram to indicate that you want to know where the line crosses the x-axis. On my diagram, I drew a big arrow pointing straight down to the top of the y-axis line.
Okay, whats next? Ah, the statements! What can you infer from the first one?
(1) The slope of line L is 3 times its y-intercept
Its tough to put this one on the diagramhow would you draw it? This is actually a really big clue for you.
Your whole goal is to try to figure out whether theres just one way to draw the line or more than one. Because this is data sufficiency, try to disprove the statement: that is, try to find more than one y-intercept that is acceptable. If so, then the statement is not sufficient.
Lets see. Say the y-intercept is 1. Then the slope would be 3. Is that allowed? Sure! The problem doesn't set any limits for the value of the slope.
What if the y-intercept is 2? Then the slope would be 6. Is that allowed? Yep, for the same reason as above.
Boom. Thats two possible values for the y-intercept, so the statement is insufficient. Eliminate answers (A) and (D) and move on to the second statement.
(2) The x-intercept of line L is [pmath]-1/3[/pmath]
Cool, a concrete piece of information. Put it on your diagram. Okay, now where could the y-intercept be?
Oh. Anywhere! Theres no information at all about the rest of the line, including the y-intercept. Not sufficient! Cross off answer (B).
Okay, here comes the tricky part: put the two statements together.
(1) The slope of line L is 3 times its y-intercept(2) The x-intercept of line L is [pmath]-1/3[/pmath]
Quick! Whats your initial instinct, right now? If you had to guess immediately, without thinking about this at all, would you guess that these two piece of info will answer the question or that they wont?
In my experience, most people will think that they do. In fact, before I actually worked through the problem myself, it did sort of seem like the two statements would work together. After all, youve got one point (the x-intercept) as well as info about the slope. Shouldnt that be enough? Is there really more than one way to draw that line?
Heres the thing: I was immediately wary of that impression because I've learned through (painful!) experience that, on data sufficiency, when something feels a certain way the opposite answer is often true. So lets dig in.
Many people, if not most, will try to combine the two pieces of info algebraically. I thought of two ways to do this.
First, translate statement one into an equation. Call the slope m and the y-intercept b. The equation is m = 3b. Substitute that equation into the standard slope-intercept equation y = mx + b:
y = (3b)x + b
You have one true point, the x-intercept: ([pmath]-1/3[/pmath], 0). Plug the point in and see what you get for b:
0 = (3b)([pmath]-1/3[/pmath]) + b
0 = -b + b
0 = 0
Huh. Thats funny. If you really know your math, then youll know what this outcome means: b could be anything. Most people, though, figure that they made a mistake somewhere or that this isnt a valid way to solve.
So maybe they try this next:
[pmath]slope={rise/run}={y_2-y_1}/{x_2-x_1}[/pmath]
We have two points: ([pmath]-1/3[/pmath], 0) and (0, b) where b is the y-intercept:
[pmath]{b-0}/{0-(-1/3)}=b/{1/3}=3b[/pmath]
Hmm, so the slope equals 3bwait a second! This statement is just saying that m = 3b. Thats what statement one says by itself and you already decided thats not sufficient.
Again, many people will assume they made a mistake here, but the real answer is that youre getting this result because there are infinite possibilities for the y-intercept.
How are you going to prove that to yourself? Fall back to the try some real numbers technique that youve already been using.
So, the x-intercept is [pmath]-1/3[/pmath]. Look at your coordinate plane. Pick a value for the y-intercept. How about 1? Okay, if the y-intercept is 1, then the slope is:
[pmath]{1-0}/{0-(-1/3)}=1/{1/3}=3[/pmath]
Does that fit the equation from statement 1? Yep, the slope is 3 times bigger than the y-intercept.
What if you make the y-intercept 2? Then, the slope is:
[pmath]{2-0}/{0-(-1/3)}=2/{1/3}=6[/pmath]
Check it out. The slope, 6, is once again 3 times bigger than the y-intercept, 2. There are at least two possibilities, so youre done.
The correct answer is (E).
Key Takeaways for Better Geometry
(1) Draw! This is key for any geometry problem. There are too many moving parts; you need to keep track of everything in a clear way.
(2) On Data Sufficiency, you can try to prove the statements mathematically but unless geometry is your favorite subject, you may drive yourself a little nuts. If the problem lends itself to what we call Testing Cases (testing numbers to find different possibilities, as we did above), then go for it!
(3) Start using your 3 steps for geometry: Draw. ID the wanted element. Infer.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.