Can You Answer This GRE Practice Problem in 30 Seconds?
Earlier this week, we posted a GRE practice problem on Facebook and asked you all to give it a try. Here it is again:
Quantity A The length of the hypotenuse of a right triangle with the legs of lengths 20 and 15
Quantity B The length of the diagonal of a rectangle with a length of 24 and a width of 7
A – Quantity A is greater.
B – Quantity B is greater.
C – The two quantities are equal.
D – The relationship cannot be determined from the information given.
Many of you answered correctly (nice work!), and some of you told us that you used the Pythagorean Theorem to solve. Now, what if I told you that you should be able to answer a question like this on the GRE in 30 seconds or less (and without the Pythagorean Theorem!), allowing you to bank valuable time for more difficult questions later on in the math section? You should. You can! After all, this is a timed exam, and you only have a limited number of minutes to prove your critical thinking prowess. Using shortcuts such as the one I’m about to cover is one of the best ways to gain speed and points on the GRE math section.
As you are reading the description in each quantity, you should be drawing and labeling quick sketches on your scratch paper. Once you’ve done this, you will see that you are dealing with the hypotenuses of two RIGHT TRIANGLES. Now, here’s where the magic happens. On geometry questions dealing with right triangles, you should always check to see if the triangle given is a Pythagorean triple (one with a defined and easily memorized ratio of side lengths.)
In this problem, Quantity A, a right triangle with legs 15 and 20, is a multiple of a 3:4:5 triangle because 15 = 5 x 3 and 20 = 5 x 4. Therefore, its hypotenuse is 5 x 5 = 25. So Quantity A is 25. If you recognized the 3:4:5 right triangle here, you didn’t have to spend any time doing calculations with the Pythagorean Theorem here, and recognized that hypotenuse in, what? 10 seconds or less?
In Quantity B, you have a rectangle with a diagonal running from corner to corner. A diagonal of a rectangle divides the rectangle into two right triangles – so you should be checking for Pythagorean triples again. In this case, each triangle has legs of length 7 and 24. You may recognize the Pythagorean triple of 7:24:25. More time saved! Quantity B is 25, and the correct answer is C.
Of course, if you have a momentary lapse of memory, you can always use the Pythagorean Theorem to solve for the missing third side. But on a timed test, knowing these Pythagorean triples cold will save you time and effort. Here are the most popular ones to know for GRE test day:
You also want to be able to recognize multiples of each of these, such as the 15:20:25 (a 3:4:5 x 5) that we saw in this practice problem.
Questions about Pythagorean triples? Ask us in the comments! For more practice questions like these, check out our GRE bootcamp.