Devotees might recall that I had planned to write this next post about timelines for ACT/SAT prep. Though this is an important topic and one to which I will certainly return, I want to change gears here and look at an actual ACT problem today.
Not just any problem, however. Over several years of teaching test prep I have worked through each of the official “Bluebook” SATs and “Redbook” ACTs countless times, and the problem below is what always pops into my mind when thinking about the hardest questions I’ve come across.
This question not only looks very difficult at first glance, but requires knowledge and application of some not-so-easy math concepts. However, I think that working through this problem will demonstrate some important test-taking techniques that are applicable throughout the math section, as well as some valuable strategies for attacking questions even when you don’t really understand it.
First, without further ado, here is the question we will be looking at:
Looks pretty menacing, huh? The diagram looks tricky and the answer choices, well, they don’t exactly seem user-friendly.
Well OK, first things first, what information did they give us in the question and what are they asking us to figure out? We’ve got a right triangle inside of a circle, and we know the lengths of two of the three sides of that triangle. They are asking us for the length of arc AC, which is the section of the circle that the triangle is touching. Now, adding to the confusion is the fact that the triangle touches the circle at point C but NOT at point A (if you look closely, there’s that bit of distance between A and B there).
When I encounter questions like this, where I feel like I have no idea what to do, one important question I ask myself is “What type of math concept does this seem like?” Is this Pythagorean Theorem, Angles in a Triangle, Circumference of a Circle, etc. I know that by this point in my math career (and this will almost always be true for juniors and seniors taking the ACT) I have learned something in one of my math classes that will help me solve this problem. If I can identify what general body of math knowledge this problem is related to, then at least I have something specific I can think about.
One thing I will do is look at the answer choices! In this problem, they all have sin, cos, and tan, so now I ask myself “What do I remember about sine, cosine, and tangent?” Hopefully, somewhere in the dark recesses of your mind, you come up with:
SOH CAH TOA!
Good bet this is relevant here. All of the answers have 1/4, which I’m guessing has something to do with the two lengths of the triangle they gave me. Alright, well if I can just figure out which combination of opposite, adjacent, and hypotenuse I need, I should at least be able to get my options down to 2 of the multiple choice options.
So 4 is definitely the hypotenuse, which means tangent is out. Now I need to figure out whether side BC (the one with the length of 1) is the opposite side (in which case we need sine) or the adjacent side (cosine). To determine this I need to decide whether this problem involves angle D in the middle (which would make 1 the “opposite” side) or are we talking about angle C near the top (which would make 1 the “adjacent” side)?
Now, remember the question: they’re asking about arc length of a circle. While you might be pretty rusty with your geometry, try to think back to the fourth quarter of geometry class when you were learning about arc lengths and areas of sectors. Which seems more relevant to those problems: the size of the central angle of the circle, or the size of the “I don’t even think there’s a term for it because we never use it” random angle at the top of the triangle that’s connected to the outside of the triangle.
If you guessed central angle D, you’re right!
Now, at last, we have at least enough to narrow down our answer choices. Relative to the central angle, we know that the opposite side (O) is 1, and the hypotenuse is 4 (H). Therefore, even without really knowing how to solve any of this, we should be able to narrow our two options down to F and H, as those are the only two that use sine, which SOH CAH TOA tells us is the one that uses the Opposite side and the Hypotenuse.
It looks like this problem will require a second post to complete, so join us next time, when we find out whether (F) or (H) is the answer we’re looking for, and review the strategies and test-taking tricks that helped us get there (and will help us with other head-scratching math questions).