A. There is no due date. Well, they are due at the same time as the portfolio is due, at semester's end on Thursday, noon, December 18!
B. Solution attempts should be included in your portfolio.
C. You may work on them collaboratively, but acknowledge the collaboration, as you should any source.
D. I won't generally discuss them in class, but I'll be happy to talk with you about them outside class.
E. To get an A or A- in the course, you will need to have good solution attempts for a few of these optional problems.
F. I'll be posting more problems as the semester progresses.

1. As a class, we have reasoned that the ratio of the shortest side (shorter leg) to the longest side (hypotenuse) in a 1/6 - , 1/3 - , 1/2 - straight angle triangle (30-60-right triangle) is 1:2. What remains to know about this kind of triangle?

2. If you answered #1 above, determine some numerical information about what remains to be known about the 30-60-right triangle. You may use Euclidean geometry for this one.

3. What is the length of the altitude of an equilateral triangle? You could answer in terms of the length of the sides of the triangle, or just assume that those sides are of unit length. CLICK for definition.

4. Explain why, referring to the diagram below, when the lower left corner of the bill is folded "over" CD, that corner "lands on" the upper edge of the bill. (By folding "over" CD, I mean that CD and its extension becomes a new fold line.)

5. In a fashion similar to that in #4, explain why, referring to the diagram above, when the corner of the bill at point C is folded over FG, that corner lands on the lower edge of the bill.

6. Use two folded, and then unfolded, dollar bills to construct a regular tetrahedron. (CLICK for definition.) Make the tetrahedron so that there are no "flaps are hanging out."

7. In class we made some visual observations about the interior angles of an equilateral triangle (equal-sided, three-angled shape). Suppose we try to proceed by analogy from three to four and consider an equal-sided, four-angled shape. Could you make a visual conclusion about the interior angles of such a shape - a conclusion similar to that for the equal-sided, three-angled shape? Justify your conclusion with explanation/example.

8. Does such a shape as the one described in #7 have a name? Comment on some essential differences between three-sided and four-sided shapes.

9. Explain why, under the initial directions for a daisy design with just one radius, the final two circles (the circles centered at points E and F) intersect in a single point, G, on the first circle.

10. In Lab 7 we used Sketchpad to construct the tangent envelope of a parabola. To review that process: A point is allowed to be anywhere on a fixed line (or segment), and, as it moves, a constructed line traces out the tangent envelope. (In folding terms, many points on the edge of a square of paper are folded onto a fixed point and the creases form the tangent envelope.)

Show that there is a point on the fold line (red), so that its distance from the fixed point is equal to its distance from the fixed line (blue). (This shows that the collection of such points forms a parabola by the standard two-dimensional definition of a parabola: The set of points equidistant from a fixed point, called the focus, and a fixed line, called the directrix.) Hint: Use the fact that the fold line is a certain perpendicular bisector and create an isosceles triangle. HINT

11. Prove that the angle trisection folding procedure works.

12. Explain why an angle inscribed in a circle is half the central angle subtended by the same arc that subtends the inscribed angle. See Lab 15 link.

13. Justify the reasoning, on page 18, for the conclusion that: Suppose that there are two circles C[1] and C[2] and a point E on the line L joining the centers of C[1] and C[2]. If the power of E with respect to each circle is the same, then for any point P on the perpendicular to L through E, E(C[1]) = E(C[2]). I.e., provide reasons for each statement made.

14. Prove that the perpendicular bisector of any chord on a circle must pass through the center of that circle.

15. Construct a (hyperbolic) angle bisector for a given angle. (This is #3b in Lab 18.)

16. Construct a (hyperbolic) bisector for a given line segment. (This is #4b in Lab 18.)

17. Construct a (hyperbolic) line that passes through a given point and is perpendicular to a given line. (This is #5b in Lab 18.)

18. Construct a (hyperbolic) line that is perpendicular to a given line through a given point that is not on that line. (This is #6b in Lab 18.)

19. Using your own script tools or the tools in the file hyperbolic_plane_scripts.gsp, investigate the interior angle bisectors of a hyperbolic triangle. Do the three angle bisectors always meet in a single point? If so, prove that such is the case, and describe anything noteworthy about the point of intersection. If not, sketch an example of a case where those three angle bisectors do not meet in a single point. See Euclid's Book IV, Proposition 4.

20. Using your own script tools or the tools in the file hyperbolic_plane_scripts.gsp, investigate the perpendicular bisectors of a hyperbolic triangle. Do the three perpendicular bisectors always meet in a single point? If so, prove that such is the case. If not, sketch an example of a case where those three perpendicular bisectors do not meet in a single point. See Euclid's Book IV, Proposition 5.

21. Construct a number line in the hyperbolic plane. As in a metric extension of Euclidean geometry, such a line would have a point corresponding to 0, a point corresponding to 1, and then the other positive integers would be determined. Your job is to construct them using the tools of hyperbolic geometry. Instead of using the idea that every integer n+1 is the same distance from n as 1 is from zero, use congruence: the hyperbolic line segment joining the point corresponding to n+1 to the point corresponding to n must be congruent to the hyperbolic line segment joining the point corresponding to 1 to the point corresponding to 0. Hint: Use the idea of the method of constructing an equilateral triangle, except that the points you need to locate will be on the given line.

22. Argue that the process you used in #21 can be extended indefinitely, so that the "numbers," i.e., the points corresponding to the integers, never leave the hyperbolic plane.