Triangles: A FEW Rules Solve Most Problems

Say you are halfway through a GRE Quant section and a triangle shows up. Three letters at the corners, one angle marked, a side length floating nearby. You freeze. You start digging for some theorem from tenth grade. Ninety seconds later you guess and move on, a little rattled.

Here is my verdict: that question was almost certainly built on one of SIX rules. Not sixty. Six. Triangle questions feel endless because the test writers dress the same few ideas in new clothes. Learn the six, train them until they fire on sight, and most triangle questions turn into notes and simple arithmetic.

The six rules

1. The three angles add up to 180 degrees. Always. Know two angles and you know the third.
2. Equal sides sit across from equal angles, and the biggest side sits across from the biggest angle. A triangle with two equal sides is called isosceles, and its two base angles match.
3. Any side must be shorter than the other two added together, and longer than their difference. With sides 4 and 9, the third side is between 5 and 13.
4. Area is one half times base times height. The height must meet the base at a right angle. A slanted side is NOT a height.
5. In a right triangle, leg squared plus leg squared equals hypotenuse squared. The hypotenuse is the longest side, across from the right angle. Memorize 3, 4, 5 and 5, 12, 13, plus their multiples like 6, 8, 10.
6. Two special right triangles. A 45, 45, 90 triangle has sides in the ratio x, x, x√2. A 30, 60, 90 triangle has sides x, x√3, 2x, with the short side across from the 30 degree angle.

Stop reading for a second. Cover that list and say all six out loud. Could you? If you got stuck on one, THAT is where your missed points are hiding.

Rule 3 in action

A big chunk of GRE Quant is Quantitative Comparison: you get two quantities and decide which one is bigger, or whether you can even tell. Triangle facts love this format.

The special triangles pay the bills

Rule 6 shows up more than people expect, because test writers love answer choices with √3 in them. Where does the 30, 60, 90 triangle come from? Cut an equilateral triangle straight down the middle. Each half has a 30, a 60 and a 90 degree angle, and the short side is exactly half the hypotenuse.

Which way was faster for you? Notice that the estimate needed nothing but a rough √3 and a look at the choices. That is not a math move. It is a strategy move.

The mistakes I watch people make every week

1. Trusting the picture. ETS says plainly in its Quant section guide that figures are not necessarily drawn to scale. An angle that LOOKS like 90 degrees may not be. INSTEAD: redraw the figure using only the facts you are given, and treat anything you are not told as unknown.
2. Doing it in your head. You think you can hold three sides and two angles in memory. You can't, not reliably, not on question 17 of a timed section. INSTEAD: label every side and angle on your own sketch before you do any math.
3. Using a slanted side as the height. This one costs more points than any "hard" question. INSTEAD: before you use the area rule, put your pen on the height and ask: does this meet the base at a right angle?
4. Grinding radicals on the calculator. The on-screen calculator is handy, but it does not know triangle rules, and turning √3 into decimals when the choices are in √3 form wastes time. INSTEAD: keep the radical, and use the calculator only when you need to compare sizes.
5. Hunting for a seventh rule. When a question stumps you, the urge is to go find a new formula. INSTEAD: go back to the list. I'll bet the rule you needed was already on it.

How to train it so it sticks

Reading a list of rules is studying. Using them under the clock, on paper, until they are automatic is training. You need the second one.

Hidden triangles are the best training ground. A rectangle with a diagonal is two right triangles. An isosceles triangle with a line dropped from the top is two right triangles. Once you see that, plenty of "new" shapes are just Rule 5 again.

Notice what happened in all three examples. Every one is a set of instructions, the same way weird symbol questions are. And every one gets sorted by one early question: which rule is this? It works like the single question that sorts combinations and permutations. Name the rule first, and the math is short.

You have two Quant sections on the GRE General Test, each with 20 questions in 35 minutes. A triangle question you can finish cleanly in under a minute buys you time for the wordy ones. What do you want MORE: the feeling of solving it in your head, or the points?

Here is your challenge for today. Take a blank sheet. Write the six rules from memory and draw a small triangle for each. Then pull up the last 10 triangle questions you missed. Before you solve each one, write the number of the rule it uses. If you can't name the rule, that question is your study list. Redo every one of them on paper, labels and all, until you can name the rule in five seconds flat.