Labs #1-4 now appear in the Archives. Labs are listed in reverse order.

Lab 10

Problem Group 10: Investigating Triangles in the Hyperbolic Plane

From the Toolbox use the Point tool to 'draw' points, the Select tool to drag, and Script tools to 'construct' objects like lines, segments, circles, perpendiculars.

1. In the previous lab you showed that, unlike the Euclidean case, the perpendicular bisectors of a hyperbolic geometry triangle may fail to intersect. However, when they do, construct, as in Euclidean geometry, the circle with the common intersection point as a center and one of the three vertices of the triangle as a point on that circle. Make a conjecture(s) about what happens generally.

2. Investigate what happens when you construct the altitudes of a hyperbolic triangle. Compare to the Euclidean case. Can you describe when the three altitudes are concurrent?

3. Investigate what happens when you construct the medians of a hyperbolic triangle. Compare to the Euclidean case. Can you describe when the three medians are concurrent?

4. Investigate how to define an angle bisector.

Lab 9

Problem Group 9: The Disk Model of the Hyperbolic Plane

From the Toolbox use the Point tool to 'draw' points, the Select tool to drag, and Script tools to 'construct' objects like lines, segments, circles, perpendiculars.

1. Open Geometer's Sketchpad, and from the File menu, in the drive C Sketch folder, open H-disk.gsp. This sketch is set up to make it easy to construct objects in hyperbolic geometry by using the hyperbolic geometry script tools, which are found in the bottom left of the tool box. Always open H-disk.gsp to construct sketches in hyperbolic geometry.

Read the text in H-disk.gsp, which describes where to find help using hyperbolic script tools.

2. Hyperbolic line segments: Draw two points, C and D, within the disk, often called the 'Poincare¢ Disk,' in honor of the mathematician Henri Poincare¢.

From Script Tools, use the "Segment" script to construct the hyperbolic line segment CD. The procedure for using script tools is to click on the script you want, Segment in this case. Then you get what looks like a Select tool with a point at the tip. Click on the desired points with this tool.

Drag points C and D around (staying within the P-Disk!) to see the effect on CD.

Find locations for C and D where CD disappears. When this happens, the hyperbolic line segment CD is part of a diameter of the disk's boundary circle and it coincides with the Euclidean line segment CD.

Draw an additional point E and construct the hyperbolic line segments CE and DE to create the triangle CDE.

Construct the three perpendicular bisectors of the triangle CDE. Construct and save examples of cases when the three are concurrent (have a common intersection) and when they fail to intersect.

3. Try to find a way to describe the cases of triangles for which the perpendicular bisectors fail to have a common intersection point.

Lab 8

Problem Group 8: Triangle Explorations

Use the free-hand tools in the Toolbox to 'draw' and the Construct menu to 'construct.'

1. (Repeated for Convenience)Draw a line segment AB, a point P not on AB, and a parallel line L through P. Construct a random point C on L and triangle ABC.

Now construct a rectangle ABDE with side AB and segment DE on L.

a. Find relationships between triangles ABC, BDC, and AEC.

b. What is the area of triangle ABC?

2. Draw a triangle ABC, and construct the three medians. (Use your 'medians' script or create one now.)

Consider the many triangles created by the vertices of the original triangle and the medians. What can you discover about the areas of these triangles and their relations among one another?

Prove that each median divides the triangle into two pieces of equal area.

If you cut a triangular shape out of a piece of cardboard, where would it balance (in an unstable fashion) on a pin point?

3. Construct the angle bisectors through vertices B and C of triangle ABC. Construct the intersection point, D, of the two bisectors. From point D construct the perpendiculars to sides AC and BC, calling the intersections of the perpendiculars and the sides AC and BC, E and F, respectively. Prove that DE and DF are equal. (More, properly, DE and DF are congruent and their measures are equal.)

For any triangle, ABC, construct a circle to which each of the sides AC, CB, and BA are tangent.

Lab 7

Problem Group 7: Area Explorations

Use the free-hand tools in the Toolbox to 'draw' and the Construct menu to 'construct.'

1. Draw a line segment AB, a point P not on AB, and a parallel line L through P. Construct a random point C on L and triangle ABC.

Now construct a rectangle ABDE with side AB and segment DE on L.

a. Find relationships between triangles ABC, BDC, and AEC.

b. What is the area of triangle ABC?

2. Draw a triangle ABC, and construct the three medians. (Use your 'medians' script or create one now.)

Consider the many triangles created by the vertices of the original triangle and the medians. What can you discover about the areas of these triangles and their relations among each other?

Lab 6

Problem Group 6: Simulations of Elementary Folding Moves

Use the free-hand tools in the Toolbox to 'draw' and the Construct menu to 'construct.'

1. Where possible, construct with Geometer's Sketchpad the lines created by each of the current top six folding moves. If you have made this construction with Sketchpad previously, you may just cite your prior construction. The Elementary Folding Moves are: You can fold

a. a line joining any two given points

b. a line onto itself at any given point on the line (Do this one in two ways.)

c. one side of an angle onto the other side (Do this one in two ways.)

d. a line onto itself so that the line folded passes through any given point

e. a line so that any one given point folds on top of any other given point

f. a given point onto a given line, so that the line folded passes through a second given point

2. A Ladder and a Bucket and a Leaning Wall

In the paint bucket trajectory problem in Lab 5, Problem #2, you probably assumed that the high wall is perpendicular to the ground. How does the trajectory change if the wall is not perpendicular to the ground? The original problem is stated below.

A paint bucket hangs from a rung near the top of a long ladder, which is leaning against a high wall. As the ladder is pulled out from the wall along the ground, the top of the ladder slides down the wall. Use Sketchpad to simulate this process and describe the paint bucket's trajectory.

Lab 5

Problem Group 5: Simulations of Sliding Ladders, Planetary Motion, and Folding

Use the free-hand tools in the Toolbox to 'draw' and the Construct menu to 'construct.'

1. Circles and Ellipses

As you know, a circle is the set of points whose distance from a fixed point is a constant. Euclidean geometry assumes that we can construct these circles. An ellipse is the set of points the sum of whose distances from two fixed points is a constant. Using Sketchpad, draw two points, and draw a line segment to represent the constant. Then figure a way to construct the corresponding ellipse.

2. A Ladder and a Bucket

A paint bucket hangs from a rung near the top of a long ladder, which is leaning against a high wall. As the ladder is pulled out from the wall along the ground, the top of the ladder slides down the wall. Use Sketchpad to simulate this process and describe the paint bucket's trajectory.

3. Pythagorean Theorem Proof

Use Sketchpad to create a dynamic sketch of the folds we used to prove the Pythagorean Theorem.