GRE Math for People Who Hate Math: Right Triangles
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Geometry is one of the most polarizing topics on the GRE. If you think it’s great, this article isn’t for you! This set of tips and tricks is for those of us who would rather have a root canal than calculate the length of a hypotenuse.
Check out this Quantitative Comparison problem:
Quantity A: The length of side AB
Quantity B: 6
You might be tempted to start applying right triangle rules. Plugging the sides into the Pythagorean Theorem (a2 + b2 = c2), you’d find:
So, the answer is (A), correct? No! The answer is actually (D), because this problem isn’t about right triangle rules. It doesn’t actually specify that ABC is a right triangle, even though it looks like one. It’s really trying to test you on a totally different triangle rule, the rule that governs the possible side lengths of a triangle.
According to that rule, every side of a triangle must be shorter than the other two sides put together. Otherwise, you’d end up with a triangle that couldn’t ‘close’:
The only rule here is that AB must be shorter than 9, and longer than 1. It can be longer or shorter than 6, so the correct answer is (D).
The lesson here isn’t about the math, though. It’s that even if you know all of the right triangle rules, you should also know whether those rules apply.
The most powerful right triangle rule is the Pythagorean Theorem. It always works on any right triangle. However, there are two mistakes I’ve seen geometry-hating GRE students make over and over, and I’d like you to avoid them. First, remember that in a2 + b2 = c2 , c is always the longest side. It’s always the side opposite the right angle. If your number instincts aren’t great, it’s easy to plug the two known sides into the wrong spots in the equation and calculate a value that doesn’t make sense.
Second, not every right triangle is a special right triangle. The ‘special’ right triangles — the 45-45-90 triangle and the 30-60-90 triangle — only represent a very small number of right triangles. The Pythagorean Theorem works on all right triangles, but the ‘special right triangles’ rules only work if you already know the angles (or the sides). Here are the most common situations in which those rules apply.
– If a right triangle has two equal legs, its angles are 45-45-90, and vice versa.
– If a right triangle has a hypotenuse twice as long as the shorter leg, its angles are 30-60-90, and vice versa.
Those might not look like the ‘special right triangles’ rules that you’ve memorized already. That’s because I’ve left out the parts that you can calculate using the Pythagorean Theorem, to make the rest easier to remember. Here’s the neat trick: if you don’t exactly remember the ratio of side lengths for a special right triangle, just use the Pythagorean Theorem!
You can memorize the rule that states that the third side length is:
Or, you can just plug in the two sides you know:
The same applies to the 45-45-90 triangle:
Memorize the bullet points above, which tell you how to recognize special right triangles and what you’re allowed to do with them. You can also memorize the side length ratios —
— but those are super easy to work out on your own, in case you forget.
My last piece of advice to Geometry haters is to know what you’re solving for. If you’re solving for the area of a triangle, you don’t need to know its angles. In general, you don’t even have to know the side lengths. In a right triangle, the side lengths can be used as a base and a height, but that’s just a coincidence.
If you’re solving for an angle, you don’t need to know the side lengths. Instead, start by applying the rules you know about angles: the sum of the three angles in any triangle is 180 degrees. The only exceptions are the special right triangles, which have a specific relationship between side lengths and angles.
Finally, if you’re solving for a side length, look for right triangles! If you find a right triangle, use the Pythagorean Theorem. Ignore the other angles, unless it’s a special right triangle. And if you can’t find or create a right triangle, consider using different rules, like the one from the first problem in this article.
In short:
- Don’t use right triangle rules unless you’re sure the triangle has a right angle!
- The most powerful right triangle rule is the Pythagorean Theorem, but it’s only useful for finding side lengths.
- The only time that side lengths and angles are related on the GRE, is when you’re handling a special right triangle. Memorize the super-easy versions of the special right triangle rules shown above, to help you recall when and how to use them.
- If you’re solving for one thing — an area, a length, or an angle — focus on rules that address that. Don’t get hung up on angle rules if you’re trying to find the length of a hypotenuse.
- Don’t panic! Solving a Geometry problem isn’t a magic trick. It’s just a series of inferences. If you make them carefully and one at a time, and keep your scratch work neat, you can conquer GRE Geometry with style.
For comprehensive guidance on all things GRE Geometry, check out our Geometry GRE Strategy Guide. 📝
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Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.