MA341 HOMEWORK #5
Due Friday, October 15

Exam reminder: Our second exam will be on Wednesday, October 20. The material covered will be everything since the last exam through this homework assignment. Regarding material in the book: You should learn and understand the definitions in Sections 2.6, 2.7, and 2.8.

1. Use Geometer's Sketchpad to construct a regular hexagon starting with the center and one vertex (corner) of the hexagon. You may use only the following constructions: Line segments joining two given points, circles with given center and point on the circle, and points of intersection of two circles. Include a brief written explanation of your steps. If you are unable to get access to Geometer's Sketchpad, you may do this construction with an actual compass and straightedge!
2. Suppose and are two points in . Suppose that a is a real number such that . Let .
   1. Prove that .
   2. Prove that .
3. Finish getting the coordinates of the icosahedron by finding those twelve points on the sides of the octahedron. Suggestion: the previous homework problem above shows how to get an arbitrary point on the line segment between two given points, using a parameter a. Do this for all twelve points using the same parameter a, and then determine for what value of a the resulting shape will consist of equilateral triangles. Make a good diagram and clearly label points and their coordinates. Do NOT use decimal approximations; use exact values (which will involve square roots). Extra Credit: Use your coordinates to display the icosahedron using Maple. The square root function in Maple is sqrt(). To display it with proper proportions, use the command polygonplot3d(icosahedron, scaling=constrained).
4. Suppose R is an rectangle, where a>1. Let S be the rectangle remaining when a square is cut off of the end of R. Assume the ratio of the length of S to the width of S is the same as the ratio of the length of R to the width of R (i.e., the rectangles are proportional).
   1. Find a.
   2. What does a have to do with the icosahedron?
5. Propose a topic for your video presentation. The presentation itself will be 3-5 minutes long per person.

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