Pythagorean theorem | Geometry | Quantitative reasoning

The Pythagorean theorem is an equation you can use to find a missing side length in a right triangle. If you know the lengths of two of the three sides, the Pythagorean theorem lets you calculate the third.

Definitions

Pythagorean theorem: An equation describing the sides of any right triangle

$a^{2} + b^{2} = c^{2}$

3 4 x (3-4-5) pythagorean triple right triangle

To solve for a missing side when you’re given the other two, substitute the known side lengths into the formula and solve:

$a^{2} + b^{2} 3^{2} + 4^{2} 9 + 16 255 = c^{2} = c^{2} = c^{2} = c^{2} = c$

In the Pythagorean theorem, $c$ represents the hypotenuse, which is the longest side of a right triangle. A problem might label the sides differently (for example, the hypotenuse might be called $x$ in a diagram), but the relationship $a^{2} + b^{2} = c^{2}$ still applies as long as you identify which side is the hypotenuse.

Pythagorean triplets

Pythagorean triplets are special sets of whole-number side lengths for right triangles. Memorizing a few common triplets can save time because you can recognize the side lengths without doing the full calculation.

Remember, these Pythagorean Triplets will always result in a right triangle! If the triangle in your question is not a right triangle, you won’t be able to use a Pythagorean Triplet.

There are many Pythagorean Triples, with the most common being:

$3 : 4 : 5$
$5 : 12 : 13$
$7 : 24 : 25$

These three sets are the most common Pythagorean Triplets used in the GRE. You should memorize all three so that you can recognize if you are presented with any two of the three numbers.

5 x 13 (5-12-13) pythagorean triple right triangle

From the list of common triplets, the side lengths in the figure match the $5 : 12 : 13$ triplet. That means the missing side length must be $x = 12$ .

A Pythagorean Triplet may be double, triple, or any multiple of the size of the original ratio. For example, a right triangle with side lengths $6$ , $8$ , and $10$ is simply a $3 : 4 : 5$ triangle that has doubled in size. For example, here are the doubled versions of the most common triplets listed above:

$6 : 8 : 10$
$10 : 24 : 26$
$14 : 48 : 50$