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:

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.