Pythagorean theorem | Geometry | Quantitative reasoning

The Pythagorean Theorem is an equation used to find the missing side length of any right triangle. In other words, if you have a right triangle with two of the three sides, you can use the Pythagorean Theorem to find the missing side length.

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 given the two other sides, you simply replace the side lengths with the variables in the problem:

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

The hypotenuse (the longest side) of any right triangle is represented by the $c$ in the Pythagorean Theorem. The question might name the sides differently, like how $x$ is actually $c$ in the image above, but that doesn’t change the equation.

Pythagorean triplets

Pythagorean Triplets are special ratios of the sides of right triangles that you can memorize in order to save time instead of using the Pythagorean Theorem.

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

After looking at the Pythagorean Triplets chart, you should notice that it matches the $5 : 12 : 13$ triple, meaning the value of $x$ in the figure must be $12$ .

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:

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