MTH 250 Graded Assignment 7, part two
Experiment with hyperbolic geometry
Plotting and measuring: Non-Euclid applet at
http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html
You can set the background to white under View>>Background. This may work better for copying/printing.
Note: Applet has printing problems, and copy/paste is not available. Grr. The trick to doing this is using the "Print Screen"
key- usually in the upper right of your keyboard. "Print Screen" (or "PrtScr" or similar) copies what is on your screen to the
clipboard, then you can paste as usual into Word. Holding down Alt+PrtScr generally will copy just the active window (try it
when you're on the Non-Euclid window).
Q1: Start off by drawing and measuring a few triangles.
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Select Constructions >> Plot Point, and plot 3 points by clicking.
Select Constructions >> Draw Segment, and click endpoints to connect (A to B, B to C, C to A).
Select Measurements >> Measure Triangle, and it will give all three side lengths and angle measures.
Select Edit >> Move Point, drag the vertices of your triangle around, and watch the measurements change. The
angle sum appears to always be less than ???
Q2: Draw a pair of parallel lines by constructing alternate interior angles equal. You can select File >> New to clear the
previous construction
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Constructions >> Draw Infinite Line. Click on two spots on
the disk, and you'll get points and a line.
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Constructions>>Draw Ray at Specific Angle.
Enter a value for the angle (default is 45)
Click B, then A, then somewhere on the disk.
You'll get ray AC and ray AB forming an angle.
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Constructions>>Draw Ray at Specific Angle.
Keep the same value as above.
Click A, then C, then somewhere on the disk, to the side of
ray AC which is opposite B. (You want alternate angles).
You should get a ray CD.
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You may need to extend ray CD into an infinite line by
choosing Constructions >> Draw Infinite Line, and clicking
points C and D.
The lines should be non-intersecting; i.e. parallel.
Measurements >> Measure Angle and measure angles
BAC and DCA to verify that alternate interior angles are
equal.
* print or print screen / paste this image*
Now, Edit >> Move point
Slide point C around. The angles should stay fixed by
construction. Can you ever get line CD to intersect line
AB, or are they always parallel? (Being the same line
doesn't count!)
Q3: A parallel to a parallel may not be parallel!
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Draw infinite line AB.
Constructions >> Draw Perpendicular to line AB through
point B by first clicking point B, then the line.
Same thing through AB through point B.
You may need to Edit>>Move Point a bit to bring points C
and D closer. Get something that looks basically like the
image to the right.
Draw a perpendicular to line AD through point D (which
will give you a point E). Because alternate interior angles
are equal (both 90), line AB and line DE are _______?
Also draw a perpendicular to line BC through point C
(which will give you a point F). Because alternate interior
angles are equal (both 90), line AB and line CF are
_______?
Go to Edit >> Move Point, and slide points C and D
around. You should be able to get things so that line DE
and line CF intersect. This means that (1) you have two
distinct lines through the same point (the point of
intersection) that are parallel to a given line, and
therefore (2) lines which are parallel to a third line are
not necessarily ___________?
* print or print screen / paste this image*
* print or print screen / paste this image*
Q4: Triangle area:
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Plot 3 points (A, B, and C).
Draw segments A to B, B to C, C to A.
Measure triangle ABC (gives all sides and all angles).
Record segment lengths:
AB =
BC =
CA =
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Draw perpendicular to segment AB through C.
Measure distance CD
CD =
Calculate value of .5(base)(height), using AB as the base,
and CD as the height:
.5(AB)(CD) =
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Draw perpendicular to segment BC through A.
Measure distance AE
AE =
Calculate value of .5(base)(height), using BC as the base,
and AE as the height:
.5(BC)(AE) =
You should recognize .5(base)(height) as the formula for the area of a triangle in Euclidean geometry. Does it work as the
area formula in hyperbolic geometry? Explain why/why not.
Q5: SAS congruence
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Start by constructing two segments of the same length:
Constructions>> Draw Segment of Specific Length.
Then, with vertices at points A (on one segment) and C
(on the other), Constructions>> Draw Ray at a Specific
Angle (pick an angle). [ Click B to A to somewhere, then D
to C to somewhere.]
By default, it should put point on those rays that are the
same length as the original segments. (You can measure
distance to check, and move point if needed.)
This means SAS = SAS by construction.
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Form triangles by drawing segment EB and segment FD.
Measure triangle ABE and triangle CDF.
Are the triangles congruent?
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Since ASA congruence can be proven from SAS
congruence, we should have that as well. Construct a pair
of triangles with ASA = ASA (construct a matching side for
each, then construct matching pairs of angles at each
endpoint). Measure. Congruent? (Hope so!)
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* print or print screen / paste this image*
* print or print screen / paste this image*