The ultimate conversion of a square into right angle pyramid

Question

This is a follow up of other puzzles. Here a general case of which the other cases are a subset.

Given a square of any size, cut it into four pieces to be reassembled into a right angle pyramid (the apex above one of the base vertices).

One of the cuts must be through the point E that may be positioned any where on the side DC.

Answer 1

We can do it as follows

Let the length of the side of the square be $1$ and $|DE| = x$.
Pick the point $E'$ on side $CB$ such that $|CE'| = x$.
Make the three straight line cuts between $E$ and $E'$, $E'$ and $A$, $A$ and $E$ as shown in the following diagram
Then the fours triangles can be rearranged to form the faces of a right-angled pyramid.
In Cartesian coordinates in $\mathbb{R}^3$ we can set the vertices of this pyramid at the points $(0,0,0)$, $(1,0,0)$, $(0,x,0)$ and $(0,0,1-x)$.