smc

The triangle can be seen as having the base on the $x$ axis and height $|5\sin \theta |$. The length of the base is $10$, thus the height must be $2$. The equation $|\sin \theta |=\frac{2}{5}$ has $\boxed{4}$ solutions, one in each quadrant. Visually, the set of points of the form $(5\cos \theta ,5\sin \theta )$ is a circle centered at $(0,0)$ with radius 5. The missing vertex of the triangle must lie on this circle. At the same time, its distance from the $x$ axis must be 2. The set of all such points are precisely the lines $y=2$ and $y=-2$, and each of these lines intersects the circle in two points.