Consider all of the possibilities of generating a triangle with three diagonals and/or sides of a regular hexagon. In each case, find the probability that a point inside the hexagon is also inside the triangle. Explain each solution.
Consider all of the possibilities of generating a triangle with three diagonals and/or sides of a regular hexagon. In each case, find the probability that a point inside the hexagon is also inside the triangle. Explain each solution.
You have correctly identified the three types of triangles in a hexagon.
Now, you must find the number of each type is possible. (do not over-count)
The area a regular hexagon is $\displaystyle A=\frac{3\sqrt{3}}{2}\ell^2$ where $\displaystyle \ell$ is the length of a side in the hexagon.
Next, you need to find the area of each type of triangle.
You are looking for the ratio of the area of the triangles to area of the hexagon.
If you know complex variables the set of numbers $\displaystyle v_k = \exp \left( {\frac{{k\pi i}}{3}} \right),\,k = 0,1, \cdots ,5$ form a hexagon inscribed in the unit circle. From that you can use the semi-perimeter rule to find the area of each of the three types of triangles.
If you don't know complex variables, I don't know how to help you further.