One application made possible by the Pythagorean identity is determining location by using a method called triangulation. Rightly named as this method requires you to create a right triangle for reference.

By finding angles to an object from known points on a predetermined baseline, one can find the distance or height of an object.. This became so crucial in the days of exploration and cartography when explorers were making representations of the earth as accurate as possible, in fact, mapmaking and surveying would not be possible if it were not for the discovery of this relationship. (ScienceOnline)

While there are situations that it may be easier to determine the distance simply by measuring it, more often than not, directly measuring the distance is unrealistic and impractical.

One of these situations in which the method of triangulation can be applied is in astronomy. Astronomists, such as the Chinese astronomists mentioned earlier, understood this as well in the past and employed such methods to determine the distance of the moon from the earth, the radius of the sun, and many other discoveries.

Another situation in which measuring a distance is crucial yet unrealistic to directly measure is in warfare. Information on distance to an object is invaluable as it could save many lives from tragedy.

In a more modern context, the technology that we use for GPS coordinates is based on methods of triangulation. Every time you use GPS, the object you are using communicates with satellites in orbit to determine exactly where you are on the earth. Again, this is an important technology that saves lives.

Here is an applet designed to help with visualizing how triangulation can be used: