Let me try a similar triangle: (0,0), (10,0), (0,10).
I assume that points on the side of the triangle are not "inside" it, so looking at each vertical line that is an integer value of x, I have x = 1, where the inside points go from y = 1 up to y = 8. We don't include y=9 because it is on the line itself, not inside.
The next value of x is x=2 where the possible integer values of y are y=1 up to y=7.
So you can see that we can just add up the numbers:
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
to answer the question.
Another way to finish the problem is to use a "summation formula". To add all the natural numbers up to n, we just calculate n(n+1) / 2. So for 1+...+8 we have 8(8+1)/2 = 36.
Now you try your question using this model.
Stephen La Rocque.