Geometry: An Interactive Journey to Mastery

Solutions

Lesson 33

1. ,WFDQDOZD\VEHGRQH7KHVROXWLRQVWRWKHQH[WIRXUSUREOHPVH[SODLQZK\


2. 3XVKWKHZRRGHQVSRRQEHWZHHQWKHJDSRQWKHRWKHUVLGHRIWKHFURVVLQJ
   as shown in Figure S.33.1.
   This result shows that we can always convert undercrossings
   to overcrossings, and vice versa, as convenient.


3. 8VH3UREOHPWRFRQYHUWRQHRIWKHFURVVLQJVIURPDQ
   undercrossing to an overcrossing, or vice versa. Then, it is clear
   WKDWWKHVWUDQGVFDQEHXQWDQJOHG௘6HHFigure S.33.2௘ͽ


4. 8VLQJWKHUHVXOWRI3UREOHPZHFDQWUDQVIRUPDQ\
   XQGHUFURVVLQJVRURYHUFURVVLQJVVRWKDWWKHEUDLGLVHTXLYDOHQWWR
   MXVWDƒURWDWLRQRIWKHEDVH௘6HHFigure S.33.3௘ͽ


5. )RUWKUHHVWUDQGVOHW³/ ́GHQRWHWKHDFWLRQRIFURVVLQJWKHOHIW
   WZRVWUDQGVDQG³5 ́GHQRWHWKHDFWLRQRIFURVVLQJWKHULJKWWZR
   VWUDQGVͼ௘%\3UREOHPLWGRHVQRWPDWWHULIWKHVHDUHXQGHUFURVVLQJVRURYHUFURVVLQJV௘ͽ7KHQDQ\EUDLG
   \RXFUHDWHFDQEHHQFRGHGDVDOLVWRIOHWWHUV
   LLLRRRLRLLRRRLLLRRLLLLLRRRRRRRLLRLRLRLLLRRRL,
   IRUH[DPSOH
   %\3UREOHPDQ\WZRFRQVHFXWLYH/VRUWZRFRQVHFXWLYH5VFDQEHGHOHWHG7KXVDQ\EUDLGLVSK\VLFDOO\
   equivalent to one of the form LRLRLRLRLRL... or RLRLRLRLR....
   %\3UREOHPDQ\WKUHHFRQVHFXWLYHWHUPVRIWKHIRUP/5/RU5/5FDQEHGHOHWHG$OLWWOHWKRXJKWVKRZV
   WKDWWKLVPHDQVWKDWWKHEUDLGLVHTXLYDOHQWWRRQHRIWKHIROORZLQJ/5/55/RUWKHEUDLGZLWKQR
   FURVVLQJVZKDWVRHYHUͼ௘WKHXQWDQJOHGVWDWH௘ͽ
   <RXFDQFKHFNWKDWQRQHRI/5/5RU5/KDYHWKHPLGGOHVWUDQGLQWKHPLGGOHSRVLWLRQ%HFDXVHWKH
   TXHVWLRQGHPDQGHGWKLVWKHRQO\RSWLRQWKDWUHPDLQVLVWKDWWKHRULJLQDOEUDLGLVSK\VLFDOO\HTXLYDOHQWWRWKH
   untangled state.

Figure S.33.1

Figure S.33.2

Figure S.33.3