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  • Proof in a Data Sufficiency Geometry Problem

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    GMAT MYTH: You need to actually solve a problem to get the correct answer.

    FACT: So many students get caught up in the idea that one must solve a problem to get the correct answer. Well, that is not the case on many questions on the GMAT. Sometimes it actually slows you down to fully work out a problem.

    Let's go to the board to check out my point.

    In the rectangle coordinate system (not shown), triangle ABC has a vertex at point (0, 56). If point B is at the origin, then how many points on line AC have integer values for both their x and y values?
    (1) The third vertex of triangle ABC lies on the x-axis, and the triangle has an area of 196.
    (2) Point A has a positive x coordinate and a y coordinate of zero.

    Here's another coordinate geometry data sufficiency question, but this time the question is a mouthful. Naturally, since there's no picture provided, you're going to be better off drawing as much of the picture as you can. The question tells us one point is at (0,56) and that a second point, B, is at (0,0). The question then asks how many points on the mysterious third line have integer values for both the x- and y-coordinates. Conceptually, it is sometimes difficult to wrap your brain around, but let's be patient.

    Statement 1 tells us the third point lies on the x-axis. We already know the first point is on the y-axis and the second is at the origin. So we are definitely looking at a right triangle. Since we are also given the area in Statement 1, we can take into account that the height of ABC is 56 and we can certainly find what the base is (remember A = ½ bh so, in turn, what the exact coordinates are of that last point). That means we know the exact dimensions of this triangle.

    Be very careful here. Do we really need to know exactly how many points have integer values for x and for y? NO!!! Regardless of whether you mark the third point on the positive side of the y-axis or the negative, you will have a definite number of points that DO NOT NEED TO BE COUNTED.

    This is a time management blunder if you attempt to try and work this out. I always tell myself to get over my ego when I am in a time pressured environment and I have what I am looking for already. There is no need to take another step on this particular portion of the problem so move to the next part.

    Also, let's get in the habit of eliminating answer choices on our scratch paper now that Statement 1 is sufficient.

    AD BCE

    Statement 2 says that Point A lies on the x-axis, but where? Statement 1 was sufficient because that third point was locked on our figure. Point A in Statement 2 slides on this coordinate plane and can create different numbers of integer values from different locations on the x-axis. If you draw Point A at (56,0), you'll get a large number of integer values; however, if you draw Point A at (2,0), you'll get only one. Statement 2 is insufficient. The answer is A.

    Bottom line: Work on your content as well as your strategies and approaches to questions.

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