Hello I need help with problem 3a.

terminology, his result is essentially our above table of the ssible finite groups of rotations (proper and improper) in two nensions." roblems Show that the 3-dimensional cube ([0, 1]3 C R3) can be sliced by planes to obtain a square, an equilateral triangle, and a regular hexagon. Prove that any two rotations Re(p) and Re(q) with the same angle 0 E R are conjugate in Iso (R2); that is, Re(q) = Tvo Re(p) . T-v, where u is a vector from p to q. (a) Let Sn, n 2 3, denote the side length of Pn, the regular n-sided polygon inscribed in the unit circle. Show that San 12 - 14 - Sm. Deduce from this that $4 = V2, S8 = V2 - V2, $16 = V2 - V2 + 12,.... Generalize these to show that S2n = 12 - V2 + V 2+ ... + V2 , with n - 1 nested square roots. MacBook Pro esc # $ % & @ 4 5 6 8 9 2 39. Symmetries of Regular Polygons (b) Let An be the area of Pn. Derive the formula Anti = 20-1s2n = 2n-1\\/2 -V2+V2+...+ v2, with n - 1 nested square roots. (Hint: Half of the sides of Pn serve as heights that of the 2n isosceles triangles that make up Pan, so that Azn = nsn/2.) Conclude I lim 2\\/ 2 - V2 + V2+ ... + V2= 1, where in the limit there are n nested square roots. In particular, we have lim V2 + V 2+ ... + V2 = 2. n -+ 0o 4. (a) Prove that in the product of three reflections, one can always arrange that one of the reflecting lines is perpendicular to both the others. (b) Derive Theorem 4 without the "orbit argument," using the previously proved fact that all the rotations in G have the same center. 5. The reciprocal of a point p = (a, b) E R2 to the circle with center at the origin and radius r > 0 is the line given by the equation ax + by = r2. Show that the reciprocals of the vertices z(2kx), k = 0, ..., n - 1, of the regular n-sided polygon Pn to its inscribed circle give the sides of another regular n-sided polygon whose vertices are the midpoints of the sides of Pn