**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.
**

**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!****Suppose and are two points in . Suppose that***a*is a real number such that . Let .- Prove that .
- Prove that .

**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).**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).- Find
*a*. - What does
*a*have to do with the icosahedron?

- Find
**Propose a topic for your video presentation. The presentation itself will be 3-5 minutes long per person.**

Thu Oct 7 13:17:52 EDT 1999