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Assumptions and Estimation


Geometry, assumption and estimation. On the GMAT, what are we allowed to assume from figures, and what are we not allowed to assume? First of all, we are always allowed to assume that lines that look straight, actually are straight. So suppose we're given this diagram, it certainly looks as if we go from B to A to C and what we have is a straight line.

Well, technically, if someone wanted to fool us instead of being exactly 180 degrees, the angle A could be, let's say 179.9 degrees and it would look straight to us but there would be a slight bend at A. We can rest assured that the GMAT is not going to play that trick. If something looks like a straight line on the GMAT that is money in the bank it is absolutely a straight line.

So, that's one thing you can trust a bit. If it looks straight, it is straight and that's really good news. Nevertheless, you cannot always trust how things look. As it turns out, you can't trust most things about diagrams. For example, if two lengths look the same but no other information is given, then you cannot just assume that they are the same.

If two lines look parallel or perpendicular, but no other information is given, then these things cannot be assumed. Check out these drawn-to-scale figures. It is very tempting to assume that what we have here is a right triangle and a square, but that's not the case. What if we find out that angle FEG is 89.9 degrees?

So that's not a right angle. In fact, triangle FEG is really an acute triangle with three angles less than 90 degrees. And if we look at what appear to be a square, and we find out that non of the angles in JKLM are right, and LM is slightly different from the other three sides, then JKLM is not even pretending to be a square.

None of the four angles are right angles, all of them are different from 90 degrees. Three of the side are same but side LM is different. So, that's nowhere near a square. We can fall into all sorts of traps if we make assumptions about figures on the GMAT. And that might beg the question, what can we trust about a figure?

Well, as it turns out on the GMAT, not a whole lot. You should be very suspicious of any figure drawn on the GMAT. The following quote appears on the GMAT official practice website. Figures are not necessarily drawn to scale. They are generalized figures showing little more than intersecting line segments and their points, angles, and regions.

For example, if a figure described as a rectangle, looks as if it might be a square, don't assume it's a square just by looking at the figure. GMAT wants us to base all our answers on geometric reasoning and deduction, not on estimation or comparing quantities by sight or measurement. So it's extremely important to appreciate all the implications of that quote. For example, suppose the GMAT gives us this diagram for a question.

Certainly, looks like a square, doesn't it? What the GMAT test writers are doing here is baiting you. They're baiting you to make the assumption that yes what you have here is a perfect square. But there is no guarantee that the figure actually is a square it just looks like one.

JKLM will be in that order, we know that for sure. And there will be four sides, we know that for sure. But the figure really could be any of the following. So, any of these could be the actual shape and the draw square is just really deceptive. The GMAT is making sure that we're not blindly trusting the shape drawn.

As a test taker, you have to realize that the shape may not be drawn to scale at all. That is very important to appreciate. So, if you're looking at that thing, that looks like a square, remember that it could be any of these four shapes you see here. All right, here is what we for sure cannot assume about GMAT figures.

We cannot assume equal lengths. We cannot assume horizontal or vertical. We cannot assume parallel or perpendicular. We cannot assume right angles, we can assume none of those things from the given figures. But if we're told that something is true in the text of the problem, then we can trust that information and use it to make further deductions about the figures.

We can't trust lengths. We can't trust angles as they've been drawn, but we can trust information given in the figure or text of the problem. So let's say that we are given this. This figure contains two valid pieces of information. In the figure above, we absolutely know that PR=8 and that angle QPR is 90 degrees.

From the given information, we can deduce that QR is the hypotenuse. So we know that QR is the longest side and therefore it must be longer than 8. We know thing that QP. QP might be less than 8, it might be equal to 8, it might be greater than 8. Those are things we cannot figure out from the diagram and of course, we would need more information in order to truly find out anything here.

In this figure, we know we have a 40 degree angle and a 50 degree angle. Now, we think about the theorem that says, well, 40 plus 50 is 90 and 180 minus 90 is 90. Three angles in a triangle have to add up to 180 degrees. So, we can deduce that angle D equals 90 degrees. We are not assuming that.

We're making a logical deduction, logical deductions you can trust. If you arrive at a dimension through logical deduction, you can absolutely trust it. So that's how we know in this particular figure that angle D is 90 degrees. It is not something we're assuming we know it as a geometric deduction. Deductions based on the rules of geometry are exactly what you should trust on the GMAT not how the figures appear.

In summary, on the GMAT, we can assume that a line that looks great is 100% straight. But that is the only thing we can determine based on appearances. Going by the figure alone, there's virtually nothing else we can assume. We must rely on facts and relationships specified in the text of the problem or the figure to make rules-based deductions.

The GMAT is tricky, getting misleading figures which tempts students into making unwarranted assumptions and falling for trap answers.

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