Download Hyperbolic Geometry

Hyperbolic Inquiry Lesson
Do d-Triangles have circumcenters?
Lesson Summary:
Students will measure distances and angles in Euclidean and Hyperbolic space on intersecting
line segments, circles and triangles to discover the character of hyperbolic space. Students will
use this knowledge to construct a triangle and determine whether triangle in hyperbolic space
have circumcenters.
Key words:
H2, unit disc, d-line, d-segment, d-circle, boundary
Background Knowledge:
Students should be familiar with Cabri software. The hyperbolic menu should be downloaded
from the internet. The lab should be completed after the students have studied the axioms and
theorems in absolute and Euclidean geometry. Specifically, students should be familiar with the
linear pair axiom, vertical pair theorem, properties of circles, properties of triangles, isosceles
triangle theorem and circumcenter of triangles.
Learning Objectives
1. To become familiar with the concept of distance and angle measure in hyperbolic space
to test axioms and theorems which are true in Euclidean space.
2. Understand the existence of perpendicular and parallel lines in hyperbolic space.
3. Determine if isosceles triangle theorem holds in hyperbolic space.
4. Use understanding of the nature of hyperbolic space to determine if hyperbolic triangles
have circumcenters.
Materials
Cabri
Access to lab via the internet
Assessment
A lab report with answers to the questions and constructions illustrating completion of the lab
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Circmcenter of d-Triangles
Lab Goal: To determine if the hyperbolic triangles have circumcenters.
Activity:
Part I. Comparing Euclidean and Hyperbolic Space (open Hyperbol.men)
Euclidean Space (this section should confirm what you already know)
1. Create two intersecting lines AB, and CD.
Label the point of intersection P.
(Use line tool)
(intersection point tool )
2. Create segment AP, PB, CP, and PD.
Hide the lines.
(Use segment tool)
(Use hide/show tool)
A
D
P
C
B
3. Measure <APD and <APC. Use the calculator function to add the angles. Record the value.
What theorem or axiom have you just illustrated? Do you predict that this will hold in
hyperbolic geometry? Why?
(use the angle measurement tool)
The angles add up and equal 180 degrees. I do not think this axiom will hold in
hyperbolic.
4. Measure <CPB. Record the value. Compare to <APD. What theorem or axiom have you just
demonstrated? Do you predict that this will hold in hyperbolic geometry? Why?
These angles are alternating interior angles. I do think this will hold in hyperbolic
geometry.
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Hyperbolic Space – The hyperbolic menus appears on the last four buttons on the toolbar.
These will be referred to as:
12. Figure Menu
13. Construction Menu
14. Reflection Menu
15. Measurement Menu
5. Create your hyperbolic plane. (On the Measurement menu – Button 15, create a unit disc by
choosing a center point and a point on the x-axis which will represent 1 unit. All constructions
made here have the properties of H2. Create a d-line (on the Figure Menu) by choosing two
points A and B and then choosing the unit circle. (Label A and B using the Euclidean label tool.)
A
D
C
0.2
0.2
B
6. Create a d-segment AB on the d-line (Figure Menu) by choosing A and B and then the unit
circle. This segment is also called an “arc”. Create d-line CD and d-segment CD such that AB
and CD intersect. Label the intersection P.
7. Measure the non-E distance of AB and CD (Using the Measurement menu, select two points,
the axis and then the unit circle). What values do you get?
AB = 2.08
and
CD = 1.82
8. Measure < APD and <APC using “angle” on the Measurement menu (On the hyperbolic
menu). Add them together. Record the value.
APD = 85.9
and
APC = 94.1
APD + APC = 180
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9. What does this suggest?
That perpendicular’s can be created in hyperbolic geometry! Sure… going with
that…
10. Measure < DPB and compare to < APC. What do you notice about the measurements?
Alternating interior angles still exist! Hyperbolic geometry sure is magical :-)
11. Move segment AB. What do you notice about the distance and angle measurements? What
does this demonstrate?
Angle BPD stays alternating interior to APC. And as you take A closer to the end of
the circle, the larger it gets.
12. Move Point B outside the circle? What happens? Why?
The length of the segment becomes non-existent since space does not exist outside of
the unit circle.
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Part II. Properties of a Circle in Hyperbolic Space
1. Create a d-circle within the unit disk. (Choose the center point first, choose a point within the
unit circle to be on the circumference, and chose the disk circle.) How does the shape of the
circle compare to that in Euclidean geometry?
It still looks like a circle, but the center of the circle is not in the same point it would
be in Euclidean.
2. Connect the center of the circle with 3 points on the d-circle using d-segments. Measure the
length of the d-segments and compare them.
The lengths of the radii are all different. I have one measuring 2.19 and another at
1.61.
3. Move the center of the d-circle toward the edge of the unit disk. What happens to the
measurements? Describe the appearance of the radii of the circle:
As the circle gets larger, the radii get longer as well. Also, when you move the center
of the d-circle to the center of the unit disk, it becomes more like a Euclidean circle.
4. What does this suggest about circles in hyperbolic space?
They do not hold the same properties as they do in Euclidean space.
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Part III. Parallel lines, Perpendicular lines and isosceles triangle theorem in hyperbolic space?
1. Parallel lines. Create d-line AB in the lower half of unit disk and a point P in the upper half
of the unit disk. Can you create more than one line that passes through P and is parallel to d-line
AB (parallel means they do not intersect)? How many lines do you think you could create?
You can create infinite lines through point P that are parallel to the d-line.
2. Perpendicular lines. Construct a perpendicular line using Euclidean circle techniques.
Construct a new unit disk with d-line AB in the lower half and point P in the upper half. Create a
d-circle such that point P is its center and it intersects d-line AB in two points other than A and
B. Label the points of intersection X and Y.
Create two d-circles such that their centers are X and Y and they both intersect point P. Mark the
other intersection between the two circles Z. Create d-segment PZ. Label the intersection
between d-segment PZ and d-line AB point W
3. Measure < PWY, < PWX, < ZWY and <ZWX using “angle” on the Measurement menu (On
the hyperbolic menu). Record your results. What do the angle measurements tell you? Is the dline PZ uniquely perpendicular to d-line AB at the point of intersection? Slide point P to
confirm.
The line is uniquely parallel. The measures for each of the listed angles is 90
degrees. As I slide P, the angles stay the same.
4. Create d-segments PY and YZ to form d-triangle PYZ. Record the non E distance of both
segments. What type of triangle have you created. Measure the base angles of the triangle. Does
the isosceles triangle theorem hold?
This is an Isosceles triangle. The theorem does not hold since the lengths of the two
sides do not have the same length, but the angles are the same.
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Part IV. D-triangles Circumcenters.
1. Recall from Euclidean geometry that the Circumcenter of a circle is the intersection of the
perpendicular bisectors of the sides of a triangle. Where is the circumcenter located in
relationship to the triangle? Do you think the same will hold for d-triangles? Why or Why not?
The circumcenter should lie inside the d-triangle. I think it will hold.
2. Construct a new unit disk. Create a d-triangle with the tool d-triangle. Construct the
perpendicular bisector for each side of the d-triangle. (use midpoint and perpendicular at a point)
Does the circumcenter exist? How can you confirm the point is the circumcenter?
The supposed circumcenter exists. We could construct the circumcircle to test to see
if this is actually the circumcenter or if circumcenters exist.
3. Create the circumcircle using d-circle. Do the perpendicular bisectors always intersect?
When they intersect does circumcircle intersect the three vertices?
The perpendicular bisectors do not always intersect, as the circumcircle grows larger
and gets closer to the edge of the unit circle, we see the perpendiculars start to split.
Yes, when they intersect, the circumcircle intersects the three vertices.
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4. Do are two perpendicular bisectors sufficient to determine the circumcenter? Why or why
not?
Yes, since two lines intersect at one point.
5. Compare and contrast the existence of the circumcenter of a triangle and a d-triangle.
The circumcircle does not always exist in a d-triangle.
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