Think of a chord as the number of vertices to advance between start and finish. So the shortest possible chord would be 1, then up to 4 they would become longer. 5 is just as long as 4, and 8 is as short as 1. So in a way, a triangle is a partition of the number 9 into 3 parts, each part being a positive integer. There are $\binom82=28$ such partitions. But in order to exclude some equilateral triangles, and also omit rotated versions of the same triangle, let's concentrate on those where the parts are in ascending order since there is always a way to read the vertex differences in non-descending order for any triangle, and a scalene triangle will not have repeated counts either.
1+2+6&=9 & 1+3+5&=9 & 2+3+4&=9
The $6$ has the same length as a $3$, so the first is indeed scalene. Likewise the $5$ has the same length as a $4$. In terms of lengths (still expresse din terms of vertices skipped) the three triangles above would correspond to $(1,2,3), (1,3,4), (2,3,4)$. All scalene as you can see.
So if you are only interested in identifying triangles up to congruence, the answer is $3$. If you consider triangles different if they get rotated or reflected, there are $9$ ways to rotate them and $2$ ways to orient them, and for scalene triangles they will all be distinct. This leads to a total count of
$$9\cdot 2\cdot 3=54$$