Write your solutions carefully and be as explicit as you can about your reasoning. I may send a proposed solution back for revision, asking you for more detail. (See diagram below.)

Go to #13

1. This first problem is really several problems in one. You may separate out and solve each by itself. (Each part will "count" as a separate problem.) We showed that by folding one edge exactly onto another edge of an equilateral triangle, you get another triangle (with two layers of paper) that has two sides in a 1:2 ratio.

a. What can you say about the angles in the "two-layer" triangle described above?

b. Justify your claims in part a by using congruency arguments.

c. What about the third side of that "two-layer" triangle? (You may use the Pythagorean Theorem to solve this one.).

2. You were asked in Lab 2 to construct, by folding, an equilateral triangle in a square. The side of the resulting triangle can be made so that it is a side of the square. (Problem 12 below asks you to prove that the folding method works.)

a. Does there exist a larger equilateral triangle triangle inside the square?

b. If you think not, justify your position. If so, can you construct it by folding?

3. How is the angle x in the figure below related to angles QPR and PRQ? Write a statement that is true about triangles, based on this observation. (You may use te Alternate Interior Angles Theorem for this one.)

4. Complete the proof of the segment trisection, following the outline in the .gsp file SegmentTrisection.gsp in the shared SummerMath network space - in the 1ExplorationsInGeometry directory/folder.

5.What is the shape of a dollar bill? Yeah, I know; it's a rectangle. But what is the shape of that rectangle, based on your experience with dollar folding? (And, assuming that the bill folds exactly as described at <http://www.mtholyoke.edu/courses/jmorrow/homework.html>, homework assignment #4.) There is a more fundamental sub-problem here: How do you describe the specifics of the shape of some rectangle, so that you distinguish it from other, non-similar, rectangles?

6-9. Complete each of the investigations, numbered 2, 3, 4, and 6, in Lab 15.

10. This problem is an extension of the segment trisection process. Roughly speaking, the idea is to repeat the process of segment trisection, but, rather than folding a corner to the midpoint of a side, fold it to the 2/3-1/3 point that we located using that trisection method. Then use the geometry of similar triangles to determine what fraction of an edge this repetition (well, let's say analogous process) produces. Below are the details:

a. Use the trisection of segment process, starting with a square and orienting it so that the lower left corner is folded to the midpoint of the top edge.

b. Mark the trisection point, which is 2/3 of the way down the right edge, with a pencil or fold.

c. Unfold and rotate the square 90 degrees counterclockwise, so the the trisection point is now on the top edge of the square.

d. Apply EFM 1 to the lower left corner and the trisection point.

e. The top edge of the folded paper intersects the right edge at a point P. How far down the right edge is this point? I.e., what fraction of the edge length of the square is this point P down the right edge?

11. Prove that the angle trisection method shown in Lab 10a works. You may use the Hint that is linked at Lab 10a.

12. This problem is an extension of #3 in Lab 7: Continue your construction, using Sketchpad to construct the creases resulting from the process of folding an equilateral triangle from a square, to produce an equilateral triangle that sits inside a square. Then use your diagram(s) as support for a proof that the method works; i.e., it actually produces a triangle that is equilateral.

13. 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.

14. Show that the Poincare disk model of the hyperbolic plane satisfies Euclid's Postulate 2. To do so, you could describe each step in the process, using the Euclidean constructions inversion, perpendicular bisector, circle, ...or make a script tool in Sketchpad for the process and submit the <.gsp> file as your solution.

15. In Lab 13 Sketchpad is used 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 construct an isosceles triangle.

The following problems are taken from Lab 18; the instructions refer to using Sketchpad.

16. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points. To make a given segment joining those points, use the segment tool. Construct another point P that is on the segment.

b. Construct a (hyperbolic) line that passes through a point P on the line and is perpendicular to the given line segment. (Hint: Mimic the Euclidean proof of Proposition 11 using the script tools in the file h-plane_constructions.gsp.)

c.Test your construction by measuring the angles formed by the given line segment and your proposed perpendicular.

d. Write a brief description of your results.

17. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw three random points: P, Q, and R. To make a given line, start with Q and R and use the line tool.

b. Construct a (hyperbolic) perpendicular to the given line through the given point P, not on that line. (Hint: Mimic the Euclidean proof of Proposition 12 using the script tools in the file h-plane_constructions.gsp.) If you follow the proof of Proposition 12, you will need to construct a fourth point (the arbitrary point D referred to in Euclid) that is on the "other" side of hyperbolic line from P.

c.Test your construction by measuring the angles formed by the given line and your proposed perpendicular.

d. Write a brief description of your results.

18. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points: P and Q. To make a given line, start with Q and R and use the line tool.

b. Now that you've constructed hyperbolic perpendiculars, how about a rectangle for which the hyperbolic line segment PQ is a side?

c. Write a brief description of your results.