GRE Special Triangles Will 30(Rock)-60-90 You
April 4, 2014
My GRE students are quite aware of my love of the show 30 Rock. I always find a way to work it into our before-class chat and even into our lessons.
That’s because 30 Rock, if you haven’t seen it, is one of the most brilliantly written shows on television. Tina Fey and her adroit team of writers weave together countless and disparate plotlines into one homogenous, hilarious conclusion. The reason I bring this up is that the very same thing happens with the quantitative content on the GRE. It’s slightly less hilarious, but it really does all come together.
Case in point: special triangles. Knowing these is a GRE lifesaver, as long as you know how to use them. The problem is, they require some prerequisite math knowledge to be used to their best advantage.
Any Kaplan student with a laminated strategy sheet can tell you that a right triangle with sides 3 and 4 in length will have a hypotenuse with the length of 5. That’s why they’re called 3-4-5 triangles. However, where my students have had trouble in the past is applying that base knowledge to save time on more advanced problems.
Say a question on the GRE asks for the perimeter of a right triangle with known side lengths of 21 and 35. You could use the Pythagorean Theorem to determine that the remaining side is 28. Then, assuming you remember to re-read the question and add all the sides together, you’d have the correct answer. You’d also have about a minute less time for the rest of the section than you would if you’d attacked the problem in a different way. To save time in the long run, sometimes you need to invest a little more time up front.
Anytime you’re working with a figure on the GRE, study it. For this problem, you see that you have a right triangle because the figure either shows that the largest angle is 90 degrees or has a box in the corner where the largest angle is. Determining you have a right triangle is your cue to start examining the side lengths to see if they fit the profile of a special triangle.
Sometimes, you’ll see a 4 and a 5, and your job is easy; the other side is 3. This special right triangle is the first you should learn – the side ratios are always 3:4:5.
Or sometimes you’ll see a side 5 and a hypotenuse 5√2 and you’ll know that the remaining side is 5 (for a 45:45:90 triangle, one of those other special triangles you’ll be sure to review).
3:4:5 Triangles in Disguise
But sometimes, what you’ll see is a special triangle in disguise. In our example, the shortest side of our right triangle was neither 3, 4 nor 5. It was 21. But 21 is a multiple of three. More specifically, it’s 3 x 7. The hypotenuse, or longest side, of the triangle was 35. 35 = 5 x 7. Now we know that we’re dealing with a 3-4-5 triangle. The remaining side has to be a multiple of 4 in the same way the other sides were multiples of 3 and 5. 4 x 7 = 28.
Now we can use the calculator to add it all together and get the correct answer, 84, for the perimeter of the triangle. Or we could save even more time by extending our knowledge of proportions and number properties.
We know that our 21-28-35 triangle was just a 3-4-5 triangle in a “times-seven” disguise. We also know that the distributive property tells us that ab + ac + ad = a(b+c+d). Doesn’t it follow then, that 3(7) + 4(7) + 5(7) = (7)(3+4+5)? On that clunky on-screen GRE calculator, you’ll find that 7 x 12 is much quicker than 21 + 28 + 35.
Now, you can get an idea of how all the content you’re reviewing for the GRE can coalesce to make your life easier. And, as you’re moving onto bigger and better things, you need this part of your life to be as easy as possible.
It’s like Jack Donaghy once said to Liz Lemon on 30 Rock: “Lemon, the grown-up dating world is like your haircut. Sometimes, awkward triangles occur.”
To do the truly grown-up thing, we must subvert the awkwardness of such triangles and appreciate them for what they are: masterful amalgams of math magic!