Triangular Tantaliser

Several pupils from The Mount School in York attempted this problem. Two pupils began to try to explain how they knew they had found all the solutions. They said:

"If you've got a base of 1 unit and a height of 1 unit then there are 3 triangles possible,

if you've a base of 1 and a height of 2 then there are another 3 possible triangles and

a base of 1 with a height of 3 gives another 3.

So you've got 9 triangles with a base of 1"

Here are two diagrams to illustrate this:

This is a good and convincing start - they made 27 triangles but do not appear to have considered triangles whose bases are not horizontal.

Can anyone develop these excellent beginnings? Perhaps the students at The Mount School could put their ideas together to come up with a more "complete" solution.