Problem 5. (20 pts) (The Torus) Let T2 = R/T be the Euclidean Torus, where T = (t(0,1); t(1,0) C Iso(R?) is the group generated by the two translations t(0,1), t(1,0) : R? + R?. We use coordinates (x, y) E T2, induced by the coordinates (x, y) E R?, where in the torus T? we identify (x, y) ~ (x +n, y +m) for any (n, m) e Z?. Let 7 : R? R?/T be the projection map, sending a point to its I-orbit. By definition, a line in the torus T2 is the image of a line L C R? under the map T. (a) Find the distance between the two points (1.1, 12), (-1.9, 23.8) E T. {x = ay} C R be the line of slope a. Suppose a is a non-zero (b) Let La rational number. Show that the intersection of 1(La) and T(Lo) consists of finitely many points. (c) Assume a is a non-zero irrational number. How many times do the lines T(La) and T(Lo) intersect ? (d) Find four lines L1, L2, L3, L4 C T such that |L; n L;| = 0 for 1