EXPLORATORY ACTIVITY III
Math 355
Objective: Given a pair of rotations ρA,Θ and ρB,Ф , identify the isometry ρB,Ф ρA,Θ .
1. Plot points A and B on a clean sheet of paper and draw the line m through A and B.
2. Construct the unique line l through A such that ρA,45 = σm σl .
3. Construct the unique line n through B such that ρB,60 = σn σm.
4. Label the point C = l ∩n.
5. Express the composition ρB,60 ρA,45 as a product of two reflections:
ρB,60 ρA,45= ________________________ .
6. The isometry in step 5 is a rotation about the point C through angle ______________.
7. Make a conjecture based on your observations in steps 1 – 6:
Conjecture 1: ___________________________________________________________.
* * * * * * * * * * * * *
8. Plot points D and E on a clean sheet of paper and draw the line m through D and E.
9. Construct the unique line l through D such that ρD,120 = σm σl .
10. Construct the unique line n through E such that ρE,240 = σn σm.
11. Explain why l and n are parallel: __________________________________________
12. Express the composition ρE,240 ρD,120 as a product of two reflections:
ρE,240 ρD,120 = ________________________ .
13. The isometry in step 12 is a ________________________.
14. If necessary, generalize the hypotheses of Conjecture 1 to include steps 8 – 13:
Conjecture 2: ____________________________________________________________
_______________________________________________________________________ .
