Why do you need a toolbox? Start with a ray. A line is simply defined by the equation:
R(t) = R0 + R*t
where R0 is a point on the line where it starts, so a vector of length 3. As well, R is a vector of length 3, that indicates the direction the ray propagates in. For any non-negative value of the parameter t, you get a point along that ray..
Next, you have a plane, thus the plane of a mirror the ray will intersect. A plane can be defined by a point in the plane, and the normal vector to the plane.
Where does the line intersect that plane? Basic analytic geometry. Once you find the point on the plane of the mirror, you can compute the distance traveled to that point.
Next, when the ray hits the mirror, what angle will it reflect at? Again, basic geometry. At this point, you have the same problem you had before. A ray that emanates along some path from a starting point, and you know the vector that defines that path! Rinse and repeat, until the ray reaches its final destination. Since you want to compute the distance traveled, sum of the individual distances to the final point.
All of this is basic geometry, something that should have been taught in high school geometry. At the very least, you were given those tools to do so then, whether you remember them or not. But surely you can find the necessary equations online.
Of course, this being YOUR student project, I've already done more than I should do. Start writing. It was assigned to you, not to me, or to anyone else you might find on the street.
