3 Steps to Better Geometry

Stacey Koprince —  February 5, 2014 — 1 Comment

GMAT-geometryA 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 – 1/3”

My title (3 Steps to Better Geometry) is doing double-duty. First, here’s the general 3-step process for any quant problem, geometry included:

Screen Shot 2014-02-05 at 12.13.43 PM

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 don’t (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. You’ll 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. You’re 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 you’re trying to “prove.”

Let’s dive into this problem. They’re talking about a coordinate plane, so you know the first step: draw a coordinate plane on your scrap paper. The question indicates that there’s a line L, but you don’t know anything else about it, so you can’t 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 wouldn’t 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, what’s next? Ah, the statements! What can you infer from the first one?

“(1) The slope of line L is 3 times its y-intercept”

It’s tough to put this one on the diagram—how would you draw it? This is actually a really big clue for you.

Your whole goal is to try to figure out whether there’s 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.

Let’s 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. That’s 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 – 1/3”

Cool, a concrete piece of information. Put it on your diagram. Okay, now where could the y-intercept be?

Oh. Anywhere! There’s 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 – 1/3”

Quick! What’s 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 won’t?

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, you’ve got one point (the x-intercept) as well as info about the slope. Shouldn’t that be enough? Is there really more than one way to draw that line?

Here’s 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 let’s 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: ( – 1/3, 0). Plug the point in and see what you get for b:

0 = (3b)(- 1/3) + b
0 = -b + b
0 = 0

Huh. That’s funny. If you really know your math, then you’ll know what this outcome means: b could be anything. Most people, though, figure that they made a mistake somewhere or that this isn’t a valid way to solve.

So maybe they try this next:

Screen Shot 2014-02-05 at 12.19.21 PM

We have two points: ( – 1/3 , 0) and (0, b) where b is the y-intercept:

Screen Shot 2014-02-05 at 12.20.22 PM

Hmm, so the slope equals 3b…wait a second! This statement is just saying that m = 3b. That’s what statement one says by itself and you already decided that’s not sufficient.

Again, many people will assume they made a mistake here, but the real answer is that you’re 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 you’ve already been using.

So, the x-intercept is – 1/3. 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:

Screen Shot 2014-02-05 at 12.21.27 PM

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:

Screen Shot 2014-02-05 at 12.22.15 PM

Check it out. The slope, 6, is once again 3 times bigger than the y-intercept, 2. There are at least two possibilities, so you’re 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.

Stacey Koprince

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Stacey Koprince is an Instructor and Trainer as well as the Director of Online Community for Manhattan Prep. She's also a management consultant who specializes in corporate strategy. She has been teaching various standardized tests for more than fifteen years and her entire teaching philosophy can be summed up in five words: teaching students how to think.

One response to 3 Steps to Better Geometry

  1. I hate geometry it is hard for me to understand but your explanation is very detailed and gave me a better idea what i didn’t get before. You’re way better than my teacher in high school, I feel like I’m at school again.

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