Math 444 Assignment 8A (Due Monday, 11-25, more to follow in 8B)

Read Carefully Brown, Section 2.4: The Algebra of Rotations.  This expands on the class work last Friday.

8-1 On a separate sheet you will be given a page with a segment AB.  Construct with straightedge and compass a regular pentagon ABCDE with this segment as a side. (We will do this in class.)

8-2 Given a quadrilateral ABCD with angle ABC = angle BCD and AB = CD, prove

a)      AD is parallel to BC

b)      ABCD is a cyclic quadrilateral (i.e., it can be inscribed in a circle).

c)      Use this to show that any regular n-gon can be inscribed in a circle. (A regular n-gon defined as a polygon with all n sides congruent and all n angles congruent)

8-3 On a separate page, you will be given a segment OA.  Construct a regular pentagon ABCDE so that O is the center of the circle circumscribed about the pentagon.

8-4 Let A and B be points in the plane and let A₇₂ and B₇₂ be rotations by 72 degrees centered at each of these points.  Then the composition B₇₂A₇₂ is a rotation with center C by angle c.  (See the reading in Brown and the class handout from Friday.)

a)      Tell what is angle c. (Numerical answer)

b)      Construct with straightedge and compass the point C.

c)      What is the ratio AC/AB (exact numerical answer)?

8-1 Construct a regular pentagon ABCDE with this side AB.

8-3 Construct with straightedge and compass a regular pentagon ABCDE with this radius OA of the circumscribed circle.