Some people would choose to diagram this question. Let’s work it out mathematically first, and then we’ll see what a diagram might look like.
If we’re thinking in terms of the Pythagorean theorem, the distance between the two points will be the hypotenuse in the theorem, and the horizontal and vertical changes between the two points will be the legs in the theorem.
We know that the hypotenuse, or the distance, is 17.
We know that the separation in the y axis is 15, because point B has a y value of 18 and point A has a y value of 3, and 18 - 3 is 15. We’re looking for one possible value of x, so we need to know how big the separation in the x-axis must be in order for it to be true that a2 + b2 = c2 when a is 15 and c is 17. So let’s figure it out. 152 is 225, and 172 is 289. So that gives us this:
225 + b2 = 289 (Pythagorean theorem)
b2 = 64 (combine like terms)
b = 8 or -8 (isolate b)
So we know that the separation in the x-axis must be 8. (Don’t make the mistake of thinking that 8 is the answer to the question! 8 is just the length of the separation in one axis, but the question is asking for an actual coordinate in the x-axis.)
Since the x-axis separation is 8 units and the x-value for B is 10, we know that the x-value for point A must be either 2 or 18. So the answer can be either 2 or 18.
Now let’s take a look at what we might get if we decide to diagram this. First, let’s get a rough idea of where point B is:
When we try to plot point A, we start to see the difficulty in the question: we have one coordinate for A, but not
the other, so we can’t plot it. All we can do is say that A must lie somewhere on the line of y = 3, because its y-value is 3.
So now the issue is that A is 17 units away from B, and A is also somewhere on the dashed line in the diagram. So we have to figure out exactly where A can go. At this point, I would draw a circle (or part of a circle) showing the points that are 17 units away from B. A must be on a point where this circle intersects the line y = 3 above.
Now, all we have to do is figure out the x-value of either one of those two spots where A could be. For this, we’ll want to use the Pythagorean theorem. The separation between the y-values of the two points will be one leg in our right triangle. The separation between the x-values of the two points will be the other leg. And the hypotenuse will be the 17-unit straight-line distance between the two points:
From here, we can apply the Pythagorean theorem as we did in the other approach, and realize that the horizontal leg in the triangle must have a length of 8 units, which means that the x-value for point A must be either 2 or 18.
Again, we could have done this in a more formal way by using the distance formula:
d = √((y2 - y1)2 + (x2 - x1)2 )
But, in general, I advise you to avoid formulaic thinking on the SAT, because there are so many instances on the average SAT Math section in which a formula might seem appropriate when it actually isn’t. And even when a formula might work on a particular question, it’s almost always going to take more time and energy than a non-formulaic approach, and it will also typically increase the likelihood of a mistake on your part.
Also, in the case of the distance formula, I find that it’s much easier for most people to remember and apply the Pythagorean theorem than to recall every detail of the distance formula correctly. And since the SAT doesn’t care how you arrive at the answer, I generally advise people to work in whichever way seems easiest at the time.
Page 671, Question 15
For this question, the key thing is understanding what the idea of modeling the data means. We're looking for the line that provides the best match to the general trend of the data points. Since the data doesn't increase or decrease as you move from left to right, the only thing that works is (A).
Note that (B), (C), and (D) would all have to go through the point (0,0), which clearly isn’t appropriate here. Lastly, (E) would be increasing as it went to the right—it would pass through (0,44) and (5,49). We don’t want that kind of positive slope because if we look at the plotted points we can see that the values are not increasing overall as we move left-to-right. The highest value is in the second trial, and one of the lowest values is in the 5th.
This is one more example of a question that’s fairly easy if we read carefully and pay attention, but that many test-takers will miss anyway.
Page 671, Question 16
As with many SAT Math questions, there are a lot of ways to go about this one. The easiest approach, I think, is to convert either 12L or 10L to W, and then figure out the number of L x W rectangles, since that’s what the question asks for.
The only way to convert L to W is to notice that 2L = 3W, which we know because 2L and 3W both correspond to the height of the big rectangle in the diagram.