I am trying to create a problem for my students where they are essentially finding the length of a 3D vector. I wanted the problem to only use whole numbers/perfect squares.
In the problem they are given a rectangular prism where they need to use Pythagoras' Theorem to find the diagonal of the base and they are then to use this measurement and the height to find the diagonal of the whole prism. What dimensions can I make the prism so that its side lengths, and both the diagonal of the base and the diagonal of the prism are all whole numbers?
There are an infinite number of these triple. All that is needed is to replace the hypotenuse of one triple with the odd leg of a larger triple. This, plus the even leg of he larger triple forms a Pythagorean quadruple such as: $$(3,4,5)+(5,12,13)\longrightarrow(2,4,12,13)$$
The process can be continued indefinitely since the odd leg can be any odd value greater than one, e.g.
$$3,4,12,84,3612,6526884,6526885)$$ $$ (33,56,65)+(65,72,97)\\ +(97,4704,4705)+(4705,442728,442753)\\ =(33,56,72,4704,442728,442753) $$