# GRE Geometry - Pythagorean's Theorem

Pythagorean's Theorem exploits very important characteristics of a right triangle. You should be familiar with the Pythagorean's Theorem because it will allow you answer quickly GRE geometry questions that ask you about the lengths of the sides of a right triangle. Visually, given a right triangle, with sides *x*, *y*, and *z*:

Pythagorean's Theorem is the following relationship:

*x*^{2} + *y*^{2} = *z*^{2}

Note that *z* is the **hypotenuse** of the triangle, and *x* and *y* are the legs of the right triangle.

If you "plug in" the values for *x*, *y* and *z* into the equation, you get 3^{2}+4^{2}=5^{2}, which simplifies to 9+16=25. Sounds simple enough, right? Good. Because it is.

So how can you use this relationship, or how might a geometry question on the GRE exam require you to apply Pythagorean's Theorem? Assume that you were given the above triangle, except that you were given only the values for *x* and *z*, and instead you were asked to calculate the value of *y*. This is how you would go about answering such a question:

- Write out the original question:
*x*^{2}+*y*^{2}=*z*^{2} - "Plug in" the know values:
3
^{2}+*y*^{2}=5^{2} - Simplify as much as you can:
9+
*y*^{2}=25 - Subtract 9 from both sides of the equation:
*y*^{2}=16

At which point, the question has been changed so that it asks you to determine, "what value squared yields 16?" The answer is, 4.