How many right angle triangles does a grid of dots have?

There are $\left( \frac{7\cdot8}{2}\right) ^{2} = 3136$ right triangles with vertical/horizontal legs.

There are $276\cdot 4 = 1104$ with legs of slope $\pm 1$

There are $85 \cdot 8 = 680$ with a leg of slope $\pm \frac{1}{2}$

There are $31 \cdot 8 = 248$ with a leg slope of $\pm \frac{1}{3}$

There are $12 \cdot 8 = 96$ with a leg slope of $\pm \frac{1}{4}$

At this point, I stopped since the running total is $5264$ which is $\boxed{\text{More than 5200}}$

To get the exact total by this method, I'd still need to find more slopes: $\frac{2}{3}, \frac{1}{5}, \frac{2}{5},$ etc.