So, I trust by now you know what’s going on with regular triangles, and with angles in general. Right triangles get a post all to themselves because they’re special, and have some rules of their very own. Let’s dig in, shall we?
Ancient Greece was awesome.
First, let’s briefly review the Pythagorean theorem. You know this, yes?
I trust that you do. It is, after all, basically the most important thing to have come out of Ancient Greece. Now show me what you can do with it:
Note: Figure not drawn to scale.
- In the figure above, AC = 6, BC = 10, and CM = 2√13. If N is the midpoint of AC, what is BM + MN?
(A) 3 + √13
(C) 5 + 3√2
(E) 4 + 5√3
So let’s start by filling in what we know (that’s how you should start basically every geometry problem you ever do, btw). Red markings are things we’re given, green lines are what we want to figure out:
Note that we have 2 out of 3 sides of two different right triangles: ABC and ACM. Let’s Pythagorize (not a word) them both:
ABC: 62 + AB2 = 102
36 + AB2 = 100
AB2 = 64
AB = 8
ACM: 62 + AM2 = (2√13)2
36 + AM2 = 52
AM2 = 16
AM = 4
OK, we’ve got BM now…it’s 8 – 4 = 4. Easy. How do we find MN? Help us Pythagoras, you’re our only hope!
32 + 42 = MN2
9 + 16 = MN2
25 = MN2
5 = MN
So our answer is 5 + 4 = 9. That’s choice (D). Ain’t no thang.
Special Right Triangles
Now, are you ready for some amazing news? Even though you should absolutely know the Pythagorean theorem inside out, you actually don’t have to use it very often on the SAT provided you know the 4 special right triangles. FOUR? But they only give me two at the beginning of each section! Hell yes, son. I’mma give you some extra ones. You’re welcome.
These are the two you’re given at the beginning of every math section: the 45°-45°-90° (AKA isosceles right) and the 30°-60°-90° triangles. Know these ratios cold…you’ll need them. Note: my diagrams above and below are not to scale.
…And these are the two you aren’t given on the SAT. There’s nothing worth saying about the angles; don’t worry about them. What’s important are the sides. Because the SAT aims to be a “calculator optional” test, it has a strong predilection towards “easy” numbers. There aren’t that many* sets of integers that work nicely in the Pythagorean theorem with each other, but these two (and all their multiples) do.
We call these Pythagorean Triples, and you’ll be seeing a TON of them on the SAT. Note that we saw the 3-4-5 (and its big brother the 6-8-10) in the example problem above. It’s not necessary to know these, but quick recognition of them will save you 30 seconds of work old-Greek-guy work. This is the kind of thing I’m always talking about: pattern recognition and the use of that knowledge to excel at this test.
And that’s it! That’s really all you need to know about right triangles.
* Note: there are actually a bunch of Pythagorean triples. See a partial list here. But for the purposes of the calculator optional SAT, you needn’t worry about all of them. I’ve just highlighted the ones that appear on the test commonly.