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Any time there is a right triangle. The rule of a2+b2=c2 applies to the two legs squared = the hypotenuse.
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When adding two vectors at right angles is the resultant of the vectors the algebraic sum of the two vectors?

No. Vectors add at rightangle bythe pythagoran theorem: resultant sum = square root of (vector 1 squared + vector 2 squared)


When does the Pythagorean theorem apply?

With right angle triangles


What shape does the pythagorean theorem apply to?

A right angle triangle.


What other things are related to the pythagorean theorem like the pythagoran triple?

Related to the Pythagorean theorem are Pythagorean triples, which are sets of three positive integers (a, b, c) that satisfy the equation (a^2 + b^2 = c^2). Additionally, the theorem is foundational in trigonometry, where it relates to the sine and cosine functions in right triangles. The concept of distance in the Cartesian coordinate system also derives from the Pythagorean theorem, as it calculates the distance between two points. Lastly, generalizations like the Law of Cosines extend these principles to non-right triangles.


What was Fermat's original proof of his last theorem?

Fermat's last theorem says there does not exist three positive integers a, b, and c which can satisfy the equation an + bn = cn for any integer value of n greater than 2. (2 with be pythagoran triples so we don't include that) Fermat proved the case for n=4, but did not leave a general proof. The proof of this theorem came in 1995. Taylor and Wiles proved it but the math they used was not even known when Fermat was alive so he could not have done a similar proof.