# We’re Expecting… And It’s Triplets!

The Ancient Greeks made substantial contributions to the modern world. They massively expanded our understanding of science and medicine. They made significant contributions to drama, literature and the arts. And if my history books are to be believed they also had some pretty epic facial hair. But, some of the Greeks’ biggest contributions were in the field of math. One of the great thinkers of that time was Pythagoras who along with coming up with a very important theorem for your GMAT was no slouch in the facial hair department. Let’s take a look at that theorem:

`$\dpi{120}a^2+b^2=c^2$`

This Pythagorean Theorem gives us the relationship between the lengths of the sides of a right triangle where a and b are the legs and c is the hypotenuse. It’s a formula that you’ve probably used countless times, but it has one key limitation. Since plugging in values for the legs gives you , you must take a square root in order to find c. With legs of 7 and 9 you get a hypotenuse of 11.402. With legs of 14 and 11 you get a hypotenuse of 17.804. Such “ugly” numbers are exactly why you won’t see legs of that length on the GMAT.

So what will you see instead? Triplets. There are certain combinations of leg lengths that will result in an integer as the hypotenuse. Since the GMAT is a multiple-choice test, these triplets of leg : leg : hypotenuse ratios that result in three integers are very common.

`The Triplets$\dpi{120} 3x:4x:5x$$\dpi{120} 5x:12x:13x$`

Memorizing these triplets will allow you to save vital time on the test. If you see a hypotenuse with length of 5 and one leg with length of 4, you can know instantly that the other leg has a length of 3. You may also notice that I’ve included an x next to the terms in the ratios. That’s to further remind you that they are ratios! For instance, if you have side lengths of 10 and 24, that just tells you that you’re using the second triplet with all terms multiplied by two, so the hypotenuse must be 26.

My general rule of thumb is that if you have to perform a difficult Pythagorean calculation, you’ve either missed a triplet or you’ve done something wrong. The Pythagorean theorem is a great tool, but make sure that you’re spending your time thinking about how to use it, rather than scratching out calculations by hand.