GRE Math : Pythagorean Triples Save Time

Triangles are commonly tested on the GRE because they present many shortcuts for the savvy test taker. The makers of the GRE know that the average student will rely on formulas and calculations, losing valuable time and possibly making careless mistakes. However, students who consistently report high math scores know the following tips and tricks for saving time and attacking triangles.

In a right triangle, the side opposite the right angle is called the hypotenuse, and usually labeled c for the convenience of formulas. The other two sides, called legs, are labeled a and b.

If you know the lengths of any two sides of a right triangle, you can find the length of the third side using the Pythagorean Theorem:

a²+b²=c²

Pythagorean triples are groups of three certain positive integers that fit perfectly into the Pythagorean Theorem. The most common Pythagorean triple is (3, 4, 5):

3²+4²=5²
9 + 16 = 25

Therefore, given a right triangle with a leg length of 3 and a hypotenuse of 5, you know the remaining side length is 4 without having to run the numbers through the Pythagorean Theorem:

Knowing combinations of Pythagorean triples can help you save time and potential errors by avoiding the Pythagorean Theorem. The following sets of Pythagorean triples, especially those with smaller integers, may appear on the GRE:

(3, 4, 5)	(5, 12, 13)	(8, 15, 17)	(7, 24, 25)
(9, 40, 41)	(11, 60, 61)	(12, 35, 37)	(20, 21, 29)

Note that the multiples of a triple also work in the Pythagorean Theorem. For example, multiply the (3, 4, 5) triple by 2 to get (6, 8, 10):

6²+8²=10²

You should also memorize some of the multiples of the first few Pythagorean triples:

(6, 8, 10)	(9, 12, 15)	(12, 16, 20)	(15, 20, 25)
(18, 24, 30)	(10, 24, 26)	(16, 30, 34)	(14, 48, 50)

Try to solve the following question without using a pencil and paper or a calculator:

Answer:

Pythagorean Triples: (D)

The triangle is a (5, 12, 13), so the missing side is 12. The area of a square is the length of a side squared, so .