Uday from Pate's Grammar School in the UK explained how to find all of the possible whole number side lengths:

We know that,  (base $\times$ height) ÷ 2 is the area of a triangle so we know that the height $\times$ base should equal 18. From there, we know that the factors of 18 are 1, 18, 2, 9, 3 and 6. Now we know the height and base measurements of the triangles. We use that to find the triangles.

Yash from Tanglin Trust School in Singapore checked that these can be used to form all of the possible isosceles triangles with integer coordinates:

Since isosceles triangles have two equal sides, it must have a base of a length divisible by 2 in order to have integer coordinates.

Ci Hui Minh Ngoc Ong from Kelvin Grove State College (Brisbane) in Australia found all of the different ways these triangles could be positioned on the coordinate grid. Ci Hui Minh Ngoc Ong concluded that each triangle can be positioned in 12 different ways, so a total of 36 isosceles triangles with area 9 and a vertex at (20,20) can be drawn on a coordinate grid.

Click here to see Ci Hui Minh Ngoc Ong's work.