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M4005 Geometry |
March 29, 2006 |
Handout on transformations and isometries. (Note: This is an edited version of a the handout given out in class on Monday and Wednesday.)
In class, we experimented with folding paper to visualize the Three-Reflection Theorem (your textbook, page 62).
See the handoutsused in class: 1) a rotation and 2) a glide reflection.
Hand in Friday March 31. Exercises 4.5, 4.6, 4.7 and 4.8 from the second handout on coordinates. Use the diagram here to illustrate your argument in 4.5 and 4.6.
I have given a lot of advice on how to do these problems in class. One thing that I did not warn you about is that we need to stretch the meaning of the word parallel just slightly. In the following theorems, we will say that lines l and m (in the plane) are parallel if they have no points in common or if they are the same line. As you can easily check, the facts about parallels that are used in the following---mainly theorems about transverals---remain true with this slightly more general meaning.
For 4.5: Assume that A (respectively, A') has smaller x-coordinate that B (respectively, B'). (4.1 allows us to make this assumption.) Then we can use the diagram that I provided. Assume lines AB and A'B' are parallel. Draw lines as in the diagram. Use facts about parallel lines to conclude that triangles ABC and A'B'C' are similar, where C (repectively C') is the point where the line through A parallel to the x-axis meets the line through B paralle to the y-axis. Explain why it follows from this that |BC| / |AC| = |B'C'| / |A'C'|. Finally, use the fact that for each coordinate system, there is a constant k such that m(A, B) = k |BC| / |AC| to conclude that m(A,B) = m(A', B').
For 4.6: The argument for 4.5 can be taken backwards. You will use the SAS criterion for triangle similarity, and from the similarity of two the traingles and the other parallelism conditions, it will follow that AB || A'B'.
For 4.7: The argument is almost exacly the same as 4.6, but the diagram looks different enough to warrant making a new picture.
For 4.8: Line AB is parallel to line BC if and only if these are the same line (since both contain B, so the only way for them to be parallel is to coincide).