MTG 3212
Homework 8
March 4, 2009
It’s all Sketchpad this week. No proofs. Enjoy.
1) Download the file “ant1.gsp”. This is the problem: An ant is walking in a straight line.
Unfortunately, his path is blocked by an enormous tree, and he needs to get around it. Our
ant is rather strange, and he would like to continue to walk on the exact same straight line
on the other side of the tree. Fortunately, our ant is a master of Euclidean constructions,
and so he can draw any line or circle, as long as it is not blocked by the tree. The question
is: how can the ant continue on his straight line without being able to draw a circle or line
through the tree?
If this is too many words for you, the problem boils down to this: Given a segment AB
and a circle O in the configuration given to you in “ant1.gsp”, construct a segment CD on
the other side of the circle such that A, B, C, D are collinear, and construct CD without
drawing any line or circle that intersects the circle O.
This is not terribly difficult. It is a precursor to a more difficult problem coming up later in
the course.
2) Download the files “Half-Plane Model.gsp” and ”Poincare Disk.gsp”. These are the two
Sketchpad pages that demonstrate our two models for hyperbolic geometry, as seen in class
on February 23.
a) In one of the models (I don’t care which), draw an equiangular triangle. Remember that
you can use the tools to draw segments in hyperbolic geometry, and then you can use the
tools to measure angles. You should make one triangle, and then move the corners around
until the angles are approximately the same. It will be difficult and annoying to make all
three angles exactly the same, but you can get them close (within a couple tenths of a
degree).
Now that you have your triangle, measure the side lengths. You should be able to demon-
strate one of our theorems from neutral geometry: if a triangle is equiangular, then it is
equilateral.
b) In one of the models (whichever one you didn’t use for part a), draw a segment and its
perpendicular bisector. Draw an arbitrary point on the bisector, and measure its distance to
the two endpoints of the original segment. You should be able to demonstrate another one
of our theorems from neutral geometry: if C is on the perpendicular bisector of AB, then
AC ∼ BC.
=
c) In the half plane model, draw a triangle and measure its area. Then move the points
around and try to make its area as big as possible. Determine what is necessary to increase
the size of the area, and determine how big you can make it.
3) LOCI!!! We love Loci!! Here is a fun locus to construct:
• Begin with a blank Sketchpad page, and draw a circle (we will call it Circle 1).
• Now draw another circle whose center is somewhere on Circle 1 and whose radius point
is somewhere NOT on Circle 1. We’ll call this Circle 2.
• Highlight the center of Circle 2 and Circle 2 itself. Go to “Construct”, and “Locus”
should be available at the bottom of the menu. Choose it, and you should get the
locus.
What you should get is a sampling of all the circles whose origin is on Circle 1, and that
pass through the radius point of Circle 2. These circles should imply a curve that you saw
in calculus. If you move Circle 1 and/or the radius point of Circle 2, the locus will change
and you will get different curves. Play with this, and identify the curves you are seeing.
4) Now do the same locus construction in the Poincare disk. You should get the hyperbolic
version of the curves you saw in 3).
5) There are many other loci that you can construct, either in Euclidean, hyperbolic Poincare
disk, or hyperbolic half-plane model. Some interesting ones that come to mind:
• Given a line ℓ and a point A not on ℓ, the locus of all perpendicular bisectors of AB,
where B is on ℓ.
• Given a circle O and a point A not on O, the locus of all perpendicular bisectors of
AB, where B is on O (you get different results if A is inside or outside O.
• Given a circle O, the locus of all tangent lines to O (looks neat in hyperbolic geometry).
• Given a circle O, the locus of all diameter lines of O (again, neat in hyperbolic geom-
etry).
• Given a line ℓ and a point A, the locus of all circles centered at a point B on ℓ passing
through A.
• Given two lines ℓ and m that intersect, and A a point on ℓ, the locus of all segments
AB, where B is on m and all AB has a predetermined length.
So, final problem, try (at least) one, and make me a pretty picture. If you make up one that
is not on the list and it looks especially nice, I’ll give you brownie points.