If you cut a square of fabric (on grain) and then cut it into two diagonally you get two half square triangles; the long side is on the bias and the short sides are with the grain. This is the type of triangle used most in patchwork, when you join two of them together to make another square the sides of that square are on grain and stable.

If you cut a square of fabric (on grain) and then cut diagonally in both directions to make four triangles they are quarter square triangles. The long side is on grain and the shorter sides are on the bias. Join two of those together to make a square and the outside edges of that square are on the bias and liable to stretch and fray.

The main use of quarter square triangles is for setting triangles with on-point designs. You join the bias short edges to the sides of the blocks, which hopefully are on grain and stable, and the longer on grain edge of the triangle is parallel to the edge or border of the quilt.

The tube method of making pairs of triangles already joined together, as shown in that video made pairs of **quarter square** triangles with bias short sides and on grain long sides which were already sewn together. To achieve true half square triangles ready joined to another triangle you have to start with bias cut strips - which excludes the use of jelly rolls.