A less straightforward way to answer this, but one that wouldn’t require knowing that perpendicular slopes are opposite reciprocals, would be to use your calculator. You could graph the original line on your calculator, and then graph each answer choice. This would let you see on your own which choice generated a perpendicular line. Of course, this would be a little time-consuming, and you’d have to make sure you didn’t key in the graphs wrong. But it could be a viable way to solve the question if you had forgotten perpendicular lines have slopes that are negative reciprocals of each other.
Finally, note that the y-intercepts don’t matter here. The question asks about perpendicular lines, and that idea only requires us to consider the slopes, not the intercepts. This question provides yet another example of a situation in which we must consider the answer choices as part of the question, rather than hoping to predict the correct answer choice completely from scratch—in this case, there are an infinite number of valid y-intercepts that could be part of a line perpendicular to the given line.
Page 415, Question 7
This question bothers a lot of people—but, as we often see on the SAT Math section, it really just comes down to basic properties and definitions. In this case, we need to know what a triangle is. (In case you’ve forgotten, it’s a closed three-sided figure.)
If we go through each choice and imagine trying to draw it, we’d see that they can all be drawn except for (E)—if you try to draw a triangle with two sides of 5 units and a third side of 10 units, you end up with just a line segment of 10 units, because in order to reach the endpoints of the long ‘side,’ the two short ‘sides’ have to open up completely to make a straight line.
We could also think of this in terms of the so-called “triangle inequality,” which says that the length of any side of a triangle must be less than the sum of the other two sides. But I deliberately wanted to talk about it without referring to that idea because, as I keep saying, it’s very important to learn to attack SAT Math in an informal way.
There’s another important element here, too. Notice that the question uses the word “EXCEPT.” I can’t tell you how many times people have missed questions like this because they overlooked that word, and ended up just choosing the first answer choice that would work under normal circumstances. This is one more example of how important it is to read everything on the SAT very, very carefully. It’s also a good example of the importance of checking over the other answer choices before you move on—if you overlook the word “EXCEPT” and choose (A) because it’s the first thing you see that works to make a triangle, you can hopefully catch your mistake if you glance through the other choices and notice that (B), (C), and (D) also work.
Page 415, Question 8
There are a ton of things we can learn from this question, especially. Please, please read this whole description and pay attention!
First, this question is rated 5 out of 5 for difficulty by the College Board. In other words, this is a question that a ton of people miss. But, as we’ll see in a moment, it only involves basic arithmetic. In fact, it involves a specific application of an idea from arithmetic that has possibly never, ever come up in a normal math class before. What it does NOT involve is any kind of formula, or any way that a calculator is likely to be helpful. Remember, this is SAT Math, and it’s likely to involve tricks and misdirection more than formulas.
I’d also like to point out, before we even get started, that this is a question in which several of our reliable answer-choice patterns would have allowed us to predict the correct answer without even seeing the question, just from the answer choices alone. I am NOT saying that you should ever try to answer a question just from the answer choices—I’m just saying that the answer choices are hinting very strongly at (C) being the right answer, if we know how to read them. The fact that so many people missed this question shows us that most people aren’t paying any attention to the answer choices, which is part of the reason why most people have a really hard time on the SAT Math section.
This question, then, is a positively classic example of the way most test-takers throw points away for no reason on the SAT Math section. People don’t miss this question because they don’t know basic arithmetic. They miss it because they don’t pay attention to details and they don’t check over the other answer choices and think about SAT patterns.
In other words, they miss this question because of a lack of SAT-specific skills. Don’t be like them.
Okay, enough preamble. Now let’s actually answer the question.