فرضیه ها و برآورد

Assumptions & Estimation

Assumption and estimation. On the GRE, what are we allowed to assume from diagrams, and what are allowed not to assume. First of all, we’re always allowed to assume that lines that look straight actually are straight. So suppose we’re given this diagram.

It certainly looks as if, as we go from b to a to c what we have is a straight line. Well technically, if someone wanted to full us, instead of being exactly 180 degrees, the angle at a could be say 179.9 degrees, and it would look straight to us, but there would be a slight bend at a. Well, we are guaranteed the GRE is not gonna play that trick. The G, if something looks like a straight line on the GRE, that is money in the bank.

It absolutely is a straight line. So that’s one thing you can trust. If it looks straight, it is straight. That’s the good news. Well as it turns out you can’t always trust how things look and most things you can’t trust about diagrams.

If two lengths look the same but no other indication is given then you can’t assume that they are the same. If two lines look parallel or perpendicular and no other indication is given, then you can’t assume these things. If angles look acute they may be obtuse or vice versa. You can’t even assume relative sizes, if one thing is drawn the same size or longer than another, you can’t assume that is true.

And in fact even if you’re told that you have diagrams drawn to scale, so these are in fact two drawn to scale diagrams. Certainly looks like what we have is a right angle and square here. It’s very tempting to assume. Yeah a right angle and yeah a square. But that’s not the case.

Even if things are drawn to scale, here I’ve drawn angle FEG to be 189.9 degrees. So that’s not a right angle. That is not a right triangle. In fact, triangle FEG is really an acute triangle with three angles less than 90 degrees. Similarly, 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 sides are the same, but side LM is different. So that is not even vaguely like a square. So even if you were guaranteed that the diagrams were drawn to scale there’s a lot that you could not visually assume.

So we can run into all kinds of trouble if we are naive in the way we interpret diagrams on the GRE. What can we trust about a GRE diagram. Well turns out, on the GRE quant section, not much at all. Be very suspicious of any diagram drawn on the GRE. The following quote appears on the GRE official guide and at the beginning of the GRE quant section on the test, geometric figures, such as lines, circle and triangles, quadrilaterals are not necessarily drawn to scale.

In other words, give up all hope of trusting that the diagram actually looks the way it is drawn. So you should not assume that quantities such as lengths or angles are as they appear. You should assume that straight lines are straight, we talked about that. That points on a line are in the order shown, and that more generally that all objects are in their relative position.

So relative positions we can assume what’s on the left, what’s on the right. But we cannot assume size, we cannot assume what’s bigger or what’s smaller, anything like that from the diagram. They want us to base all our, our answers on geometric reasoning and deduction, not on estimation or comparing quantities by sight or measurement. So that is incredibly important.

And it’s extremely important to appreciate all the implications of that paragraph. For example, suppose the GRE gives us this diagram for a question. Certainly looks like a square, doesn’t it? What the GRE is doing is they’re baiting you. They’re baiting you to make the assumption that yes, what you have there is a perfect square.

If they draw that diagram, there’s no guarantee that the shape actually looks that way. GK, J, K, L, M will be in that order, that we know, and there will be four sides, but it could be any of these. So any of these could be the right shape, the real shape. And the square drawn is really deceptive.

But the GRE is making sure that you’re not blindly trusting the shape drawn. 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. It could be any of these four shapes at the bottom. We can’t assume equal lengths, we can’t assume horizontal or vertical.

We can’t assume parallel or perpendicular. We can’t assume right angles. We can assume none of those from the diagram, but if we’re told something is true from the text of the problem, we know that, we, and we know anything we can deduce from that. So, you can trust your geometric deduction.

So we’ll be talking more about this in coming videos, what you can deduce from geometry, itself. We can’t trust lengths as angles as, as they appear but we can trust information given in the diagram. So we’re given this. That diagram actually contains two valid pieces of information.

In that diagram we absolutely know that side PR has a length of 8. And we absolutely know that QPR is a right angle. It’s a 90 degree angle because the little perpendicular sign is drawn. So, those we know. We know nothing from this diagram. We know nothing about QP.

QP might be less than 8. It might be equal to 8. It might be greater than 8. We know that QR is the hypotenuse so we can deduce that QR is the longest side, and therefore, it must be longer than 8. So, those are things we can figure out from the diagram.

And we, of course, we’d need more information in order to find anything here. Also, in this diagram, well, we know we have a 40 degree angle and a 50 angle. Now, we think about the, the theorem that says the three angles have to add up to 180 degrees. Well, 40 plus 50 is 90.

180 minus 90 is 90. So that means that the angle at D has to be a 90 degree angle also. So that is something we can deduce. So we’re not judging how does picture look. We’re making a logical deduction. Logical deduction you can trust.

If you arrive there by a logical deduction you absolutely can trust it. That’s how we know in this particular diagram. That is 90 degrees. And it is not assuming. Because we know it as a geometric deduction. Deductions from the rules of geometry are exactly what you should trust on the GRE, not how the diagrams look.

In summary, on the GRE, we can assume that a line that looks straight in fact is straight, but we cannot assume virtually anything else purely from a diagram. We must rely on the facts and relations specified in the problem text or in special symbols in the diagram. The GRE will lay traps, giving misleading diagrams and encouraging naive students to fall into unwarranted assumptions.

Do not naively trust the diagrams on the GRE. That’s the big take-away.