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Geometric Similarity from a Variable-Parts Perspective

Geometric Similarity from a Variable-Parts Perspective
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2020-01-14 22:39
TrianglePolygon
In collections The variable-parts perspective across the curriculum
This example demonstrates geometric similarity from a variable-parts perspective. Key mathematical concepts include properties of similar triangles and polygons. Using GeoGebra commands such as Intersect, Line, and Circle, dynamic exploration of the changes in similar figures is possible. Suitable for middle school and above students or teachers.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Line(A, xAxis) → f
  3. Line(A, yAxis) → g
  4. Circle(A, a) → c
  5. Intersect(c, g, 2) → C
  6. Intersect(c, f, 2) → D
  7. Line(D, g) → h
  8. Line(C, f) → i
  9. Circle(D, a) → d
  10. Circle(C, a) → e
  11. Intersect(e, g, 2) → E
  12. Intersect(d, f, 2) → F
  13. Circle(F, a) → k
  14. Circle(E, a) → p
  15. Intersect(p, g, 2) → G
  16. Intersect(k, f, 2) → H
  17. Circle(H, a) → q
  18. Circle(G, a) → r
  19. Intersect(r, g, 2) → I
  20. Intersect(q, f, 2) → J
  21. Circle(J, a) → s
  22. Circle(I, a) → t
  23. Intersect(s, f, 2) → K
  24. Intersect(t, g, 2) → L
  25. Circle(K, a) → c_1
  26. Circle(L, a) → d_1
  27. Intersect(c_1, f, 2) → M
  28. Intersect(d_1, g, 2) → N
  29. Circle(M, a) → e_1
  30. Circle(N, a) → f_1
  31. Intersect(e_1, f, 2) → O
  32. Intersect(f_1, g, 2) → P
  33. Circle(O, a) → g_1
  34. Circle(P, a) → h_1
  35. Intersect(g_1, f, 2) → Q
  36. Intersect(h_1, g, 2) → R
  37. Circle(Q, a) → k_1
  38. Circle(R, a) → p_1
  39. Intersect(p_1, g, 2) → S
  40. Intersect(k_1, f, 2) → T
  41. Circle(T, a) → q_1
  42. Circle(S, a) → r_1
  43. Intersect(r_1, g, 2) → U
  44. Intersect(q_1, f, 2) → V
  45. Circle(V, a) → s_1
  46. Circle(U, a) → t_1
  47. Intersect(s_1, f, 2) → W
  48. Circle(W, a) → c_2
  49. Intersect(t_1, g, 2) → Z
  50. Circle(Z, a) → d_2
  51. Intersect(c_2, f, 2) → A_1
  52. Intersect(d_2, g, 2) → B_1
  53. Circle(A_1, a) → e_2
  54. Circle(B_1, a) → f_2
  55. Intersect(e_2, f, 2) → C_1
  56. Intersect(f_2, g, 2) → D_1
  57. Circle(C_1, a) → g_2
  58. Circle(D_1, a) → h_2
  59. Intersect(g_2, f, 2) → E_1
  60. Intersect(h_2, g, 2) → F_1
  61. Circle(E_1, a) → k_2
  62. Circle(F_1, a) → p_2
  63. Intersect(k_2, f, 2) → G_1
  64. Intersect(p_2, g, 2) → H_1
  65. Circle(G_1, a) → q_2
  66. Circle(H_1, a) → r_2
  67. Intersect(q_2, f, 2) → I_1
  68. Intersect(r_2, g, 2) → J_1
  69. Circle(I_1, a) → s_2
  70. Circle(J_1, a) → t_2
  71. Intersect(s_2, f, 2) → K_1
  72. Intersect(t_2, g, 2) → L_1
  73. Circle(K_1, a) → c_3
  74. Circle(L_1, a) → d_3
  75. Intersect(c_3, f, 2) → M_1
  76. Intersect(d_3, g, 2) → N_1
  77. Circle(M_1, a) → e_3
  78. Circle(N_1, a) → f_3
  79. Intersect(e_3, f, 2) → O_1
  80. Intersect(f_3, g, 2) → P_1
  81. Circle(O_1, a) → g_3
  82. Circle(P_1, a) → h_3
  83. Intersect(g_3, f, 2) → Q_1
  84. Intersect(h_3, g, 2) → R_1
  85. Circle(Q_1, a) → k_3
  86. Circle(R_1, a) → p_3
  87. Intersect(k_3, f, 2) → S_1
  88. Intersect(p_3, g, 2) → T_1
  89. Circle(S_1, a) → q_3
  90. Circle(T_1, a) → r_3
  91. Intersect(q_3, f, 2) → U_1
  92. Intersect(r_3, g, 2) → V_1
  93. Circle(U_1, a) → s_3
  94. Circle(V_1, a) → t_3
  95. Intersect(s_3, f, 2) → W_1
  96. Intersect(t_3, g, 2) → Z_1
  97. Circle(W_1, a) → c_4
  98. Circle(Z_1, a) → d_4
  99. Intersect(c_4, f, 2) → A_2
  100. Intersect(d_4, g, 2) → B_2
  101. Circle(A_2, a) → e_4
  102. Circle(B_2, a) → f_4
  103. Intersect(e_4, f, 2) → C_2
  104. Intersect(f_4, g, 2) → D_2
  105. OrthogonalLine(E, g) → j
  106. OrthogonalLine(G, g) → l
  107. OrthogonalLine(I, g) → m
  108. OrthogonalLine(L, g) → n
  109. OrthogonalLine(N, g) → b
  110. OrthogonalLine(P, g) → i_1
  111. OrthogonalLine(R, g) → j_1
  112. OrthogonalLine(S, g) → l_1
  113. OrthogonalLine(U, g) → m_1
  114. OrthogonalLine(Z, g) → n_1
  115. OrthogonalLine(B_1, g) → a_1
  116. OrthogonalLine(D_1, g) → b_1
  117. OrthogonalLine(F_1, g) → i_2
  118. OrthogonalLine(H_1, g) → j_2
  119. OrthogonalLine(J_1, g) → l_2
  120. OrthogonalLine(L_1, g) → m_2
  121. OrthogonalLine(N_1, g) → n_2
  122. OrthogonalLine(P_1, g) → a_2
  123. OrthogonalLine(R_1, g) → b_2
  124. OrthogonalLine(T_1, g) → i_3
  125. OrthogonalLine(V_1, g) → j_3
  126. OrthogonalLine(Z_1, g) → l_3
  127. OrthogonalLine(B_2, g) → m_3
  128. OrthogonalLine(D_2, g) → n_3
  129. OrthogonalLine(F, f) → a_3
  130. OrthogonalLine(H, f) → b_3
  131. OrthogonalLine(J, f) → g_4
  132. OrthogonalLine(K, f) → h_4
  133. OrthogonalLine(M, f) → i_4
  134. OrthogonalLine(O, f) → j_4
  135. OrthogonalLine(Q, f) → k_4
  136. OrthogonalLine(T, f) → l_4
  137. OrthogonalLine(V, f) → m_4
  138. OrthogonalLine(W, f) → n_4
  139. OrthogonalLine(A_1, f) → p_4
  140. OrthogonalLine(C_1, f) → q_4
  141. OrthogonalLine(E_1, f) → r_4
  142. OrthogonalLine(G_1, f) → s_4
  143. OrthogonalLine(I_1, f) → t_4
  144. OrthogonalLine(K_1, f) → a_4
  145. OrthogonalLine(M_1, f) → b_4
  146. OrthogonalLine(O_1, f) → f_5
  147. OrthogonalLine(Q_1, f) → g_5
  148. OrthogonalLine(S_1, f) → h_5
  149. OrthogonalLine(U_1, f) → i_5
  150. OrthogonalLine(W_1, f) → j_5
  151. OrthogonalLine(A_2, f) → k_5
  152. OrthogonalLine(C_2, f) → l_5
  153. Intersect(h, n_3) → E_2
  154. Intersect(n_3, a_3) → F_2
  155. Intersect(n_3, b_3) → G_2
  156. Intersect(n_3, g_4) → H_2
  157. Intersect(n_3, h_4) → I_2
  158. Intersect(n_3, i_4) → J_2
  159. Intersect(n_3, j_4) → K_2
  160. Intersect(n_3, k_4) → L_2
  161. Intersect(n_3, l_4) → M_2
  162. Intersect(n_3, m_4) → N_2
  163. Intersect(n_3, n_4) → O_2
  164. Intersect(n_3, p_4) → P_2
  165. Intersect(n_3, q_4) → Q_2
  166. Intersect(n_3, r_4) → R_2
  167. Intersect(n_3, s_4) → S_2
  168. Intersect(n_3, t_4) → T_2
  169. Intersect(n_3, a_4) → U_2
  170. Intersect(b_4, n_3) → V_2
  171. Intersect(n_3, f_5) → W_2
  172. Intersect(n_3, g_5) → Z_2
  173. Intersect(n_3, h_5) → A_3
  174. Intersect(n_3, i_5) → B_3
  175. Intersect(n_3, j_5) → C_3
  176. Intersect(n_3, k_5) → D_3
  177. Intersect(n_3, l_5) → E_3
  178. Intersect(m_3, l_5) → F_3
  179. Intersect(l_3, l_5) → G_3
  180. Intersect(j_3, l_5) → H_3
  181. Intersect(i_3, l_5) → I_3
  182. Intersect(b_2, l_5) → J_3
  183. Intersect(a_2, l_5) → K_3
  184. Intersect(n_2, l_5) → L_3
  185. Intersect(m_2, l_5) → M_3
  186. Intersect(l_2, l_5) → N_3
  187. Intersect(j_2, l_5) → O_3
  188. Intersect(i_2, l_5) → P_3
  189. Intersect(b_1, l_5) → Q_3
  190. Intersect(a_1, l_5) → R_3
  191. Intersect(n_1, l_5) → S_3
  192. Intersect(m_1, l_5) → T_3
  193. Intersect(l_1, l_5) → U_3
  194. Intersect(j_1, l_5) → V_3
  195. Intersect(i_1, l_5) → W_3
  196. Intersect(b, l_5) → Z_3
  197. Intersect(n, l_5) → A_4
  198. Intersect(m, l_5) → B_4
  199. Intersect(l, l_5) → C_4
  200. Intersect(j, l_5) → D_4
  201. Intersect(i, l_5) → E_4
  202. Segment(A, D_2) → m_5
  203. Segment(D, E_2) → n_5
  204. Segment(F, F_2) → p_5
  205. Segment(H, G_2) → q_5
  206. Segment(J, H_2) → r_5
  207. Segment(K, I_2) → s_5
  208. Segment(M, J_2) → t_5
  209. Segment(O, K_2) → a_5
  210. Segment(Q, L_2) → b_5
  211. Segment(T, M_2) → c_5
  212. Segment(V, N_2) → d_5
  213. Segment(W, O_2) → e_5
  214. Segment(A_1, P_2) → f_6
  215. Segment(C_1, Q_2) → g_6
  216. Segment(E_1, R_2) → h_6
  217. Segment(G_1, S_2) → i_6
  218. Segment(I_1, T_2) → j_6
  219. Segment(K_1, U_2) → k_6
  220. Segment(M_1, V_2) → l_6
  221. Segment(O_1, W_2) → m_6
  222. Segment(Q_1, Z_2) → n_6
  223. Segment(S_1, A_3) → p_6
  224. Segment(U_1, B_3) → q_6
  225. Segment(W_1, C_3) → r_6
  226. Segment(A_2, D_3) → s_6
  227. Segment(C_2, E_3) → t_6
  228. Segment(D_2, E_3) → a_6
  229. Segment(B_2, F_3) → b_6
  230. Segment(Z_1, G_3) → c_6
  231. Segment(V_1, H_3) → d_6
  232. Segment(T_1, I_3) → e_6
  233. Segment(R_1, J_3) → f_7
  234. Segment(P_1, K_3) → g_7
  235. Segment(N_1, L_3) → h_7
  236. Segment(L_1, M_3) → i_7
  237. Segment(J_1, N_3) → j_7
  238. Segment(H_1, O_3) → k_7
  239. Segment(F_1, P_3) → l_7
  240. Segment(D_1, Q_3) → m_7
  241. Segment(B_1, R_3) → n_7
  242. Segment(Z, S_3) → p_7
  243. Segment(U, T_3) → q_7
  244. Segment(S, U_3) → r_7
  245. Segment(R, V_3) → s_7
  246. Segment(P, W_3) → t_7
  247. Segment(N, Z_3) → a_7
  248. Segment(L, A_4) → b_7
  249. Segment(I, B_4) → c_7
  250. Segment(G, C_4) → d_7
  251. Segment(E, D_4) → e_7
  252. Segment(C, E_4) → f_8
  253. Segment(A, C_2) → g_8
  254. Intersect(r_5, d_7) → F_4
  255. Polygon(A, J, F_4) → t1, f_9, a_8, j_8

AI Drawing Prompt Reference

Draw a circle c centered at point A, then draw lines h and i through points C and D respectively intersecting the y-axis and x-axis. Next, draw circles d and e with centers at points D and C and radius a. Then, draw lines j and k through points E and F intersecting the y-axis and x-axis. Continue this process to form a series of similar triangles and circles, ensuring each circle has the same radius.
The prompt above was compiled from the original drawing commands and is for reference only.