You can use pythagoras' theorem to determine the height of a mountain if you know how far away from it you are. Measure the angle from ground to tip using a protractor and plumb bob (pendulum) and pug these two results into the theorem.
You can also find the distance to nearby stars by using a telescope and observing six months apart.This gives you 2 Astronomical Units (AU) ie 2x 93 million miles since the Earth has moved to the opposite side of the solar system in that time This is he base of your triangle. Measure the difference in angle to your chosen star in January and June for example and if there is any measurable difference then it must be a near star.eg Alpha Proxima
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Yes, the corollary to one theorem can be used to prove another theorem.
The Pythagorean theorem is used today for the same thing it was invented for: to describe the relationship between the length of the three sides of a right triangle. Using the Pythagorean theorem, you can find the the length of the third side of a right triangle with two known lengths. This can be useful in a variety of math-based situations, such as when you need to determine the distance between two known points on a graph.
pythagoreans theorem was used to help build the pyramids
Today, in our modern, sophisticated, and technologically advanced times, it can be used to find the length of any side of a right triangle, if only the lengths of the other two sides are known.
Both Thévenin's theorem and Norton's theorem are used to simplify circuits, for circuit analysis.