The SAT Prep Black Book(98)

We do have a formula for the area of a circle, though. And we could use that formula to find the area of the whole circle altogether, and then divide the area of the circle up to find the area of the shaded region.

So now we have to figure out how to find the area of the circle. On the SAT, there’s only one way the College Board can ask us to do that: we have to use A = pi(r2), where A is the area and r is the radius. (Again, this formula is provided at the beginning of the math section if you don’t remember it.)

How can we find the radius? In this case, we use the only number provided anywhere in the diagram: we look at the fact that the side length of the square is 2 units, and we realize that this corresponds to the diameter of the circle. If the diameter is 2, then the radius is 1. So the radius of the circle is 1 unit, which makes the area of the entire circle pi(12), which is just pi.

Now we have to figure out what portion of the circle the shaded area represents. There are two ways to do this. Since the diagram is drawn to scale, we could just eyeball it and realize that the shaded area is one-quarter of the circle. If we want to be more precise, we could realize that the angle at point O must be a 90-degree angle, since O is the center of the circle and the center of the square. And 90 degrees is one-quarter of 360 degrees, so, again, the shaded area must be a quarter of the circle.

That means the correct answer here is pi/4. Again, pi is the area of the entire circle, and the shaded region is 1/4 of the circle, so its area is pi/4. And (A) is correct.

There are a lot of ways to mess this question up, and most of them will involve either thinking that the radius is 2 or accidentally thinking that the shaded portion is 1/2 of the circle (this can happen if a person tries to work out the area mathematically and makes a mistake in the process, instead of just looking at the picture). Notice that both mistakes are reflected in the wrong answer choices. In fact, (E) is what you’d get if you made both mistakes together.

This is a question I would definitely double-check, or even triple-check, for several reasons. The first reason is that the answer choice patterns aren’t really pointing to (A) being right, even though it is. The patterns are really suggesting that (B) would be right (it’s in the middle of a geometric series with (A) and (C) and (E), and there are more 2’s in the answer choices than 4’s). Also, even apart from the answer choices, this is a question where it would be very easy to make a simple mistake and be off by a factor of 2 or 4, and the answer choices are clearly waiting for that.

This is a great example of the kind of question that causes the most trouble for the most test-takers. It’s something the vast, vast majority of test-takers have the skills and knowledge to answer correctly, but it’s also something where a simple mistake or two can easily be made, wrecking the question. If you want to improve your SAT Math score, this kind of question is where you should probably focus your energy first. Most people would significantly increase their scores if they just stopped giving away points on questions like this, rather than focusing on questions that seem harder.





Page 415, Question 6


This question asks about perpendicular lines. We need to know that when two lines are perpendicular to one another, their slopes are opposite reciprocals (this was an idea discussed in the toolbox earlier in this book). This idea is crucial to solving this question (at least when it comes to the way most people will solve it), but notice that the idea is a property of perpendicular lines, not a formula. Remember that challenging math questions on the SAT rely much more heavily on properties and definitions than on formulas.

To solve this question in the most straightforward way, it would probably be best to get the expression for the original line into slope-intercept format (the one that looks like y = mx + b). If we do that, we get this:

x + 3y = 12

3y = -x + 12 (get the x on the right-hand side)

y = -x/3 + 4 (isolate y)

So the slope of the given line is -1/3.

A line perpendicular to this must have a slope that is the opposite reciprocal of -1/3, which is 3. So (C) must be correct, because it gives a coefficient of 3 for x in the y = mx + b format, where m is the slope.

Note that we have the usual kinds of wrong answers we’d expect on an SAT Math question. The slope of (B) is the opposite of the correct slope, and we can probably imagine how a person might make that mistake. The slope of (D) has the right sign (it’s positive) but uses 1/3 instead of 3, and, again, that’s an understandable mistake.

Notice, also, that there’s no other slope in the answer choices that has both its reciprocal and its opposite in the choices. This is a manifestation of the imitation pattern, and it’s a good sign that (C) is, indeed, correct.