Geometry: An Interactive Journey to Mastery

Solutions

2. 2QO\WKHUHJXODUSRO\JRQVZLWKDQHYHQQXPEHURIVLGHVKDYHƒURWDWLRQDOV\PPHWU\


3. 7KLVTXHVWLRQLVGLI¿FXOWWRDQVZHUEHFDXVHWKHUHDUHPDQ\V\PPHWULHV
   7KHUHDUHOLQHUHÀHFWLRQV\PPHWULHVDERXWWKHOLQHVWKDWIROORZDQHGJHRIDQ\WULDQJOHDQGDERXWOLQHVWKDW
   JRWKURXJKDYHUWH[RIDWULDQJOHDQGELVHFWLWVRSSRVLWHVLGH
   7KHUHDUHURWDWLRQDOV\PPHWULHVRIPXOWLSOHVRIƒDERXWHDFKYHUWH[LQWKH¿JXUHDQGPXOWLSOHVRIƒ
   DERXWWKHFHQWHURIHDFKWULDQJOHDQGƒDERXWWKHFHQWHURIHDFKHGJHRIHDFKWULDQJOH
   7KHUHDUHWUDQVODWLRQV\PPHWULHVLQPDQ\GLIIHUHQWGLUHFWLRQVͼ௘QRWMXVWWKHWKUHHREYLRXVGLUHFWLRQV௘ͽDQG
   there are plenty of glide symmetries.


4. L௘ͽ 'UDZDVTXDUH
   LL௘ͽ 'UDZDQRQVTXDUHUHFWDQJOH
   LLL௘ͽ 'UDZDNLWHWKDWLVQRWDUKRPEXV


5. 7KLVTXHVWLRQRIFRXUVHGHSHQGVRQWKHIRQW\RXXVHWRZULWHWKHVHOHWWHUVͼ௘)RUH[DPSOHLQRQHIRQW³: ́
   GRHVKDYHYHUWLFDOOLQHV\PPHWU\EXWQRWLQDQRWKHU௘ͽ
   D௘ͽ %&'(+,2;
   E௘ͽ $+,0279:;<
   F௘ͽ &+,'(DQG&2'(IRUH[DPSOHKDYHKRUL]RQWDOOLQHV\PPHWU\020DQG7227IRUH[DPSOHKDYH
   vertical line symmetry.
   Lesson 31


6. 7KLVIUDFWDOLVREWDLQHGIURPGLYLGLQJDVTXDUHLQWRVXEVTXDUHVUHPRYLQJRIWKHPDQGLWHUDWLQJ:H
   see that the fractal is composed of 4 scaled copies of itself, each at^13 the scale. Thus,
   14 ˜original size = ©¹§· ̈ ̧ 31 d˜ original size,
   giving 3d = 4. Thus, d | 1.26.