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Dilations and Rotation Investigation

Dilations and Rotation Investigation
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2016-05-03 09:25
PolygonTransformationGeometric Construction
This example demonstrates the transformation rules of shapes after dilation and rotation. The core math concepts involved include dilation, rotation, fractal geometry, constructions, quadrilaterals, real numbers, and integral calculus. The construction uses commands and objects such as Circle, Segment, Rotate, Ray, Point, Dilate, and Polygon in GeoGebra. Suitable for high school students learning geometric transformations and analytical geometry.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. Rotate(B, 45°, A) → B'
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Circle(A, C) → d
  7. Segment(A, C) → h
  8. Rotate(A, 45°, A) → A'
  9. Rotate(C, 45°, A) → C'
  10. Segment(A', C') → h'
  11. Rotate(B', 45°, A) → B''
  12. Segment(C', D) → i
  13. Segment(D, C) → j
  14. Dilate(C', k, A) → C''
  15. Dilate(D, k, A) → D'
  16. Segment(C'', D') → i'
  17. Dilate(D, k, A) → D'_1
  18. Dilate(C, k, A) → C'_1
  19. Segment(D'_1, C'_1) → j'
  20. Polygon(C', B'', D) → poly1, d_1, c', b''
  21. Polygon(D, B'', D', B') → poly2, d_2, b''_1, d', b'
  22. Polygon(C, D, B') → poly3, b'_1, c_1, d_3
  23. Dilate(C', k, A) → C''_1
  24. Dilate(B'', k, A) → B'''
  25. Dilate(D, k, A) → D'_2
  26. Polygon(C''_1, B''', D'_2) → poly1', d'_2, c''_1, b'''
  27. Dilate(C, k, A) → C'_2
  28. Dilate(D, k, A) → D'_3
  29. Dilate(B', k, A) → B''_1
  30. Polygon(C'_2, D'_3, B''_1) → poly3', b''_2, c'_2, d'_3
  31. Dilate(D, k, A) → D'_4
  32. Dilate(B'', k, A) → B'''_1
  33. Dilate(D', k, A) → D''
  34. Dilate(B', k, A) → B''_2
  35. Polygon(D'_4, B'''_1, D'', B''_2) → poly2', d'_4, b'''_1, d'', b''_3
  36. Rotate(C', 45°, A) → C''_2
  37. Rotate(B'', 45°, A) → B'''_2
  38. Rotate(D, 45°, A) → D'_5
  39. Polygon(C''_2, B'''_2, D'_5) → poly1'_1, d'_5, c''_2, b'''_2
  40. Rotate(C, 45°, A) → C'_3
  41. Rotate(D, 45°, A) → D'_6
  42. Rotate(B', 45°, A) → B''_3
  43. Polygon(C'_3, D'_6, B''_3) → poly3'_1, b''_4, c'_3, d'_6
  44. Rotate(D, 45°, A) → D'_7
  45. Rotate(B'', 45°, A) → B'''_3
  46. Rotate(D', 45°, A) → D''_1
  47. Rotate(B', 45°, A) → B''_4
  48. Polygon(D'_7, B'''_3, D''_1, B''_4) → poly2'_1, d'_7, b'''_3, d''_1, b''_5
  49. Rotate(C''_1, 45°, A) → E
  50. Rotate(B''', 45°, A) → F
  51. Rotate(D'_2, 45°, A) → G
  52. Polygon(E, F, G) → poly1'', g_1, e, f_1
  53. Rotate(C'_2, 45°, A) → H
  54. Rotate(D'_3, 45°, A) → I
  55. Rotate(B''_1, 45°, A) → J
  56. Polygon(H, I, J) → poly3'', j_1, h_1, i_1
  57. Rotate(D'_4, 45°, A) → K
  58. Rotate(B'''_1, 45°, A) → L
  59. Rotate(D'', 45°, A) → D'''
  60. Rotate(B''_2, 45°, A) → M
  61. Polygon(K, L, D''', M) → poly2'', k_1, l, d''', m
  62. Rotate(C''_2, 45°, A) → N
  63. Rotate(B'''_2, 45°, A) → O
  64. Rotate(D'_5, 45°, A) → P
  65. Polygon(N, O, P) → poly4, p, n, o
  66. Rotate(D'_7, 45°, A) → Q
  67. Rotate(B'''_3, 45°, A) → R
  68. Rotate(D''_1, 45°, A) → S
  69. Rotate(B''_4, 45°, A) → T
  70. Polygon(Q, R, S, T) → poly5, q, r, s, t
  71. Rotate(C'_3, 45°, A) → U
  72. Rotate(D'_6, 45°, A) → V
  73. Rotate(B''_3, 45°, A) → W
  74. Polygon(U, V, W) → poly6, w, u, v
  75. Rotate(E, 45°, A) → E'
  76. Rotate(F, 45°, A) → F'
  77. Rotate(G, 45°, A) → G'
  78. Polygon(E', F', G') → poly1''', g', e', f'
  79. Rotate(K, 45°, A) → K'
  80. Rotate(L, 45°, A) → L'
  81. Rotate(D''', 45°, A) → Z
  82. Rotate(M, 45°, A) → M'
  83. Polygon(K', L', Z, M') → poly2''', k', l', z_1, m'
  84. Rotate(H, 45°, A) → H'
  85. Rotate(I, 45°, A) → I'
  86. Rotate(J, 45°, A) → J'
  87. Polygon(H', I', J') → poly3''', j'_1, h'_1, i'_1
  88. Rotate(U, 45°, A) → U'
  89. Rotate(V, 45°, A) → V'
  90. Rotate(W, 45°, A) → W'
  91. Polygon(U', V', W') → poly6', w', u', v'
  92. Rotate(Q, 45°, A) → Q'
  93. Rotate(R, 45°, A) → R'
  94. Rotate(S, 45°, A) → S'
  95. Rotate(T, 45°, A) → T'
  96. Polygon(Q', R', S', T') → poly5', q', r', s', t'
  97. Rotate(N, 45°, A) → N'
  98. Rotate(O, 45°, A) → O'
  99. Rotate(P, 45°, A) → P'
  100. Polygon(N', O', P') → poly4', p', n', o'
  101. Rotate(E', 45°, A) → E''
  102. Rotate(F', 45°, A) → F''
  103. Segment(E'', F'') → g''
  104. Rotate(E', 45°, A) → E''_1
  105. Rotate(F', 45°, A) → F''_1
  106. Rotate(G', 45°, A) → G''
  107. Polygon(E''_1, F''_1, G'') → poly7, g''_1, e''_1, f''_1
  108. Rotate(K', 45°, A) → K''
  109. Rotate(L', 45°, A) → L''
  110. Rotate(Z, 45°, A) → Z'
  111. Rotate(M', 45°, A) → M''
  112. Polygon(K'', L'', Z', M'') → poly8, k'', l'', z', m''
  113. Rotate(H', 45°, A) → H''
  114. Rotate(I', 45°, A) → I''
  115. Rotate(J', 45°, A) → J''
  116. Polygon(H'', I'', J'') → poly9, j'', h'', i''
  117. Rotate(U', 45°, A) → U''
  118. Rotate(V', 45°, A) → V''
  119. Rotate(W', 45°, A) → W''
  120. Polygon(U'', V'', W'') → poly6'', w'', u'', v''
  121. Rotate(Q', 45°, A) → Q''
  122. Rotate(R', 45°, A) → R''
  123. Rotate(S', 45°, A) → S''
  124. Rotate(T', 45°, A) → T''
  125. Polygon(Q'', R'', S'', T'') → poly5'', q'', r'', s'', t''
  126. Rotate(N', 45°, A) → N''
  127. Rotate(O', 45°, A) → O''
  128. Rotate(P', 45°, A) → P''
  129. Polygon(N'', O'', P'') → poly4'', p'', n'', o''
  130. Rotate(H'', 45°, A) → H'''
  131. Rotate(I'', 45°, A) → I'''
  132. Rotate(J'', 45°, A) → J'''
  133. Polygon(H''', I''', J''') → poly9', j''', h''', i'''
  134. Rotate(K'', 45°, A) → K'''
  135. Rotate(L'', 45°, A) → L'''
  136. Rotate(Z', 45°, A) → Z''
  137. Rotate(M'', 45°, A) → M'''
  138. Polygon(K''', L''', Z'', M''') → poly8', k''', l''', z'', m'''
  139. Rotate(E''_1, 45°, A) → A_1
  140. Rotate(F''_1, 45°, A) → B_1
  141. Rotate(G'', 45°, A) → G'''
  142. Polygon(A_1, B_1, G''') → poly7', g''', a_1, b_1
  143. Rotate(U'', 45°, A) → U'''
  144. Rotate(V'', 45°, A) → V'''
  145. Rotate(W'', 45°, A) → W'''
  146. Polygon(U''', V''', W''') → poly6''', w''', u''', v'''
  147. Rotate(Q'', 45°, A) → Q'''
  148. Rotate(R'', 45°, A) → R'''
  149. Rotate(S'', 45°, A) → S'''
  150. Rotate(T'', 45°, A) → T'''
  151. Polygon(Q''', R''', S''', T''') → poly5''', q''', r''', s''', t'''
  152. Rotate(A_1, 45°, A) → C_1
  153. Rotate(B_1, 45°, A) → D_1
  154. Rotate(G''', 45°, A) → E_1
  155. Polygon(C_1, D_1, E_1) → poly7'', e_1, c_2, d_4
  156. Rotate(K''', 45°, A) → F_1
  157. Rotate(L''', 45°, A) → G_1
  158. Segment(F_1, G_1) → a
  159. Rotate(K''', 45°, A) → H_1
  160. Rotate(L''', 45°, A) → I_1
  161. Rotate(Z'', 45°, A) → Z'''
  162. Rotate(M''', 45°, A) → J_1
  163. Polygon(H_1, I_1, Z''', J_1) → poly8'', h_2, i_2, z''', j_2
  164. Rotate(H''', 45°, A) → K_1
  165. Rotate(I''', 45°, A) → L_1
  166. Rotate(J''', 45°, A) → M_1
  167. Polygon(K_1, L_1, M_1) → poly9'', m_1, k_2, l_1
  168. Rotate(N'', 45°, A) → N'''
  169. Rotate(O'', 45°, A) → O'''
  170. Rotate(P'', 45°, A) → P'''
  171. Polygon(N''', O''', P''') → poly4''', p''', n''', o'''
  172. Rotate(U''', 45°, A) → N_1
  173. Rotate(V''', 45°, A) → O_1
  174. Rotate(W''', 45°, A) → P_1
  175. Polygon(N_1, O_1, P_1) → poly10, p_1, n_1, o_1
  176. Rotate(Q''', 45°, A) → Q_1
  177. Rotate(R''', 45°, A) → R_1
  178. Rotate(S''', 45°, A) → S_1
  179. Rotate(T''', 45°, A) → T_1
  180. Polygon(Q_1, R_1, S_1, T_1) → poly11, q_1, r_1, s_1, t_1
  181. Rotate(N''', 45°, A) → U_1
  182. Rotate(O''', 45°, A) → V_1
  183. Rotate(P''', 45°, A) → W_1
  184. Polygon(U_1, V_1, W_1) → poly12, w_1, u_1, v_1
  185. Rotate(N_1, 45°, A) → Z_1
  186. Rotate(O_1, 45°, A) → A_2
  187. Rotate(P_1, 45°, A) → B_2
  188. Polygon(Z_1, A_2, B_2) → poly10', b_2, z_2, a_2
  189. Rotate(Q_1, 45°, A) → C_2
  190. Rotate(R_1, 45°, A) → D_2
  191. Rotate(S_1, 45°, A) → E_2
  192. Rotate(T_1, 45°, A) → F_2
  193. Polygon(C_2, D_2, E_2, F_2) → poly11', c_3, d_5, e_2, f_2
  194. Rotate(U_1, 45°, A) → G_2
  195. Rotate(V_1, 45°, A) → H_2
  196. Rotate(W_1, 45°, A) → I_2
  197. Polygon(G_2, H_2, I_2) → poly12', i_3, g_2, h_3
  198. Rotate(K_1, 45°, A) → J_2
  199. Rotate(L_1, 45°, A) → K_2
  200. Rotate(M_1, 45°, A) → L_2
  201. Polygon(J_2, K_2, L_2) → poly9''', l_2, j_3, k_3
  202. Rotate(H_1, 45°, A) → M_2
  203. Rotate(I_1, 45°, A) → N_2
  204. Rotate(Z''', 45°, A) → O_2
  205. Rotate(J_1, 45°, A) → P_2
  206. Polygon(M_2, N_2, O_2, P_2) → poly8''', m_2, n_2, o_2, p_2
  207. Rotate(C_1, 45°, A) → Q_2
  208. Rotate(D_1, 45°, A) → R_2
  209. Rotate(E_1, 45°, A) → S_2
  210. Polygon(Q_2, R_2, S_2) → poly7''', s_2, q_2, r_2
  211. Rotate(J_2, 45°, A) → T_2
  212. Rotate(K_2, 45°, A) → U_2
  213. Rotate(L_2, 45°, A) → V_2
  214. Polygon(T_2, U_2, V_2) → poly13, v_2, t_2, u_2
  215. Rotate(M_2, 45°, A) → W_2
  216. Rotate(N_2, 45°, A) → Z_2
  217. Rotate(O_2, 45°, A) → A_3
  218. Rotate(P_2, 45°, A) → B_3
  219. Polygon(W_2, Z_2, A_3, B_3) → poly14, w_2, z_3, a_3, b_3
  220. Rotate(Q_2, 45°, A) → C_3
  221. Rotate(R_2, 45°, A) → D_3
  222. Rotate(S_2, 45°, A) → E_3
  223. Polygon(C_3, D_3, E_3) → poly15, e_3, c_4, d_6
  224. Dilate(C''_1, k, A) → F_3
  225. Dilate(B''', k, A) → G_3
  226. Dilate(D'_2, k, A) → H_3
  227. Polygon(F_3, G_3, H_3) → poly1''_1, h_4, f_3, g_3
  228. Dilate(D'_4, k, A) → I_3
  229. Dilate(B'''_1, k, A) → J_3
  230. Dilate(D'', k, A) → D'''_1
  231. Dilate(B''_2, k, A) → K_3
  232. Polygon(I_3, J_3, D'''_1, K_3) → poly2''_1, i_4, j_4, d'''_1, k_4
  233. Dilate(C'_2, k, A) → L_3
  234. Dilate(D'_3, k, A) → M_3
  235. Dilate(B''_1, k, A) → N_3
  236. Polygon(L_3, M_3, N_3) → poly3''_1, n_3, l_3, m_3
  237. Dilate(F_3, k, A) → O_3
  238. Dilate(G_3, k, A) → P_3
  239. Dilate(H_3, k, A) → Q_3
  240. Polygon(O_3, P_3, Q_3) → poly16, q_3, o_3, p_3
  241. Dilate(I_3, k, A) → R_3
  242. Dilate(J_3, k, A) → S_3
  243. Dilate(D'''_1, k, A) → T_3
  244. Dilate(K_3, k, A) → U_3
  245. Polygon(R_3, S_3, T_3, U_3) → poly17, r_3, s_3, t_3, u_3
  246. Dilate(L_3, k, A) → V_3
  247. Dilate(M_3, k, A) → W_3
  248. Dilate(N_3, k, A) → Z_3
  249. Polygon(V_3, W_3, Z_3) → poly18, z_4, v_3, w_3
  250. Dilate(O_3, k, A) → A_4
  251. Dilate(P_3, k, A) → B_4
  252. Dilate(Q_3, k, A) → C_4
  253. Polygon(A_4, B_4, C_4) → poly16', c_5, a_4, b_4
  254. Dilate(R_3, k, A) → D_4
  255. Dilate(S_3, k, A) → E_4
  256. Dilate(T_3, k, A) → F_4
  257. Dilate(U_3, k, A) → G_4
  258. Polygon(D_4, E_4, F_4, G_4) → poly17', d_7, e_4, f_4, g_4
  259. Dilate(V_3, k, A) → H_4
  260. Dilate(W_3, k, A) → I_4
  261. Dilate(Z_3, k, A) → J_4
  262. Polygon(H_4, I_4, J_4) → poly18', j_5, h_5, i_5
  263. Dilate(A_4, k, A) → K_4
  264. Dilate(B_4, k, A) → L_4
  265. Dilate(C_4, k, A) → M_4
  266. Polygon(K_4, L_4, M_4) → poly16'', m_4, k_5, l_4
  267. Dilate(D_4, k, A) → N_4
  268. Dilate(E_4, k, A) → O_4
  269. Dilate(F_4, k, A) → P_4
  270. Dilate(G_4, k, A) → Q_4
  271. Polygon(N_4, O_4, P_4, Q_4) → poly17'', n_4, o_4, p_4, q_4
  272. Dilate(U_3, k, A) → R_4
  273. Dilate(H_4, k, A) → S_4
  274. Dilate(I_4, k, A) → T_4
  275. Dilate(J_4, k, A) → U_4
  276. Polygon(S_4, T_4, U_4) → poly18'', u_4, s_4, t_4