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Step 8: Geometric Construction of Vectors and Polygons

Step 8: Geometric Construction of Vectors and Polygons
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2015-12-09 07:53
PolygonVectorCoordinate GeometryCalculusGeometric Construction
In collections Hands in the Air: Square Translation Tessellation
Step 8 demonstrates the construction of a two‑dimensional polygon involving random variable vectors using GeoGebra. The example visualizes how translation and vector operations affect the shape, and it computes the median value and definite integral of the area bounded by the polygon and the plane. Key mathematical concepts include vector arithmetic, coordinate geometry, random variables, definite integrals, and planar area. The construction uses commands such as Polygon, Segment, Vector, Translate, and Circle, defining points A, B, etc., and generating objects like poly1. This material is suitable for high‑school or early university students studying vector geometry and calculus.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(B, E) → e
  3. Segment(E, F) → f
  4. Segment(F, G) → g
  5. Segment(G, H) → h
  6. Segment(H, I) → i
  7. Segment(I, J) → j
  8. Segment(J, K) → k
  9. Segment(K, L) → l
  10. Segment(L, M) → m
  11. Segment(M, N) → n
  12. Segment(N, O) → p
  13. Segment(O, P) → q
  14. Segment(P, Q) → r
  15. Segment(Q, R) → s
  16. Segment(R, S) → t
  17. Segment(S, T) → a_1
  18. Segment(T, U) → b_1
  19. Segment(U, V) → c_1
  20. Segment(V, W) → d_1
  21. Segment(W, Z) → e_1
  22. Segment(Z, A_1) → f_1
  23. Segment(A_1, B_1) → g_1
  24. Segment(B_1, C_1) → h_1
  25. Segment(C_1, C) → i_1
  26. Vector(B, A) → u
  27. Translate(B, u) → B'
  28. Translate(C, u) → C'
  29. Segment(B', C') → b'
  30. Translate(B, u) → B'_1
  31. Translate(E, u) → E'
  32. Segment(B'_1, E') → e'
  33. Translate(E, u) → E'_1
  34. Translate(F, u) → F'
  35. Segment(E'_1, F') → f'
  36. Translate(F, u) → F'_1
  37. Translate(G, u) → G'
  38. Segment(F'_1, G') → g'
  39. Translate(G, u) → G'_1
  40. Translate(H, u) → H'
  41. Segment(G'_1, H') → h'
  42. Translate(H, u) → H'_1
  43. Translate(I, u) → I'
  44. Segment(H'_1, I') → i'
  45. Translate(I, u) → I'_1
  46. Translate(J, u) → J'
  47. Segment(I'_1, J') → j'
  48. Translate(J, u) → J'_1
  49. Translate(K, u) → K'
  50. Segment(J'_1, K') → k'
  51. Translate(K, u) → K'_1
  52. Translate(L, u) → L'
  53. Segment(K'_1, L') → l'
  54. Translate(L, u) → L'_1
  55. Translate(M, u) → M'
  56. Segment(L'_1, M') → m'
  57. Translate(M, u) → M'_1
  58. Translate(N, u) → N'
  59. Segment(M'_1, N') → n'
  60. Translate(N, u) → N'_1
  61. Translate(O, u) → O'
  62. Segment(N'_1, O') → p'
  63. Translate(O, u) → O'_1
  64. Translate(P, u) → P'
  65. Segment(O'_1, P') → q'
  66. Translate(P, u) → P'_1
  67. Translate(Q, u) → Q'
  68. Segment(P'_1, Q') → r'
  69. Translate(Q, u) → Q'_1
  70. Translate(R, u) → R'
  71. Segment(Q'_1, R') → s'
  72. Translate(R, u) → R'_1
  73. Translate(S, u) → S'
  74. Segment(R'_1, S') → t'
  75. Translate(S, u) → S'_1
  76. Translate(T, u) → T'
  77. Segment(S'_1, T') → j_1
  78. Translate(T, u) → T'_1
  79. Translate(U, u) → U'
  80. Segment(T'_1, U') → k_1
  81. Translate(U, u) → U'_1
  82. Translate(V, u) → V'
  83. Segment(U'_1, V') → l_1
  84. Translate(V, u) → V'_1
  85. Translate(W, u) → W'
  86. Segment(V'_1, W') → m_1
  87. Translate(W, u) → W'_1
  88. Translate(Z, u) → Z'
  89. Segment(W'_1, Z') → n_1
  90. Translate(Z, u) → Z'_1
  91. Translate(A_1, u) → D_1
  92. Segment(Z'_1, D_1) → p_1
  93. Translate(A_1, u) → E_1
  94. Translate(B_1, u) → F_1
  95. Segment(E_1, F_1) → q_1
  96. Translate(B_1, u) → G_1
  97. Translate(C_1, u) → H_1
  98. Segment(G_1, H_1) → r_1
  99. Translate(C_1, u) → I_1
  100. Translate(C, u) → C'_1
  101. Segment(I_1, C'_1) → s_1
  102. Segment(B, J_1) → t_1
  103. Segment(J_1, K_1) → a_2
  104. Segment(K_1, L_1) → b_2
  105. Segment(L_1, M_1) → c_2
  106. Segment(M_1, N_1) → d_2
  107. Segment(N_1, O_1) → e_2
  108. Segment(O_1, P_1) → f_2
  109. Segment(P_1, Q_1) → g_2
  110. Segment(Q_1, R_1) → h_2
  111. Segment(R_1, A) → i_2
  112. Vector(A, D) → v
  113. Translate(A, v) → A'
  114. Translate(B, v) → B'_2
  115. Segment(A', B'_2) → a'
  116. Translate(B'_1, v) → S_1
  117. Translate(B, v) → B'_3
  118. Translate(J_1, v) → U_1
  119. Segment(B'_3, U_1) → j_2
  120. Translate(J_1, v) → V_1
  121. Translate(K_1, v) → W_1
  122. Segment(V_1, W_1) → k_2
  123. Translate(K_1, v) → Z_1
  124. Translate(L_1, v) → A_2
  125. Segment(Z_1, A_2) → l_2
  126. Translate(L_1, v) → B_2
  127. Translate(M_1, v) → C_2
  128. Segment(B_2, C_2) → m_2
  129. Translate(M_1, v) → D_2
  130. Translate(N_1, v) → E_2
  131. Segment(D_2, E_2) → n_2
  132. Translate(N_1, v) → F_2
  133. Translate(O_1, v) → G_2
  134. Segment(F_2, G_2) → p_2
  135. Translate(O_1, v) → H_2
  136. Translate(P_1, v) → I_2
  137. Segment(H_2, I_2) → q_2
  138. Translate(P_1, v) → J_2
  139. Translate(Q_1, v) → K_2
  140. Segment(J_2, K_2) → r_2
  141. Translate(Q_1, v) → L_2
  142. Translate(R_1, v) → M_2
  143. Segment(L_2, M_2) → s_2
  144. Translate(R_1, v) → N_2
  145. Translate(A, v) → A'_1
  146. Segment(N_2, A'_1) → t_2
  147. Translate(u, v) → u'
  148. Translate(B', v) → B''
  149. Circle(T_1, O_2) → c_3
  150. Circle(P_2, Q_2) → d_3
  151. Segment(Q_1, J') → a_3
  152. Polygon(A, R_1, Q_1, J', I', H', G', F', E') → poly2, a_4, r_3, q_3, j'_1, i'_1, h'_1, g'_1, f'_1, e'_1
  153. Polygon(R_2, S_2, T_2, U_2, V_2, W_2) → poly3, r_4, s_3, t_3, u_2, v_2, w_2
  154. Polygon(Q_1, P_1, F, G, H, I, J, K, L, M, N, O, O_1, P, Q, R, S, T, U, V, W, Z, A_1, N_1, B_1, C_1, C, U_1, W_1, A_2, C_2, E_2, G_2, I_2, M_1, K_2, M_2, A', H_1, F_1, D_1, Z', W', V', T', L_1, S', R', Q', P', O', N', M', L', K', J', K_1, J_1, B, E) → poly4, q_4, e_3, f_3, g_3, h_3, i_3, j_4, k_4, l_4, m_4, n_4, p_3, o, p_4, q_5, r_5, s_4, t_4, u_1, v_1, w, z_1, o_1, a_5, b_4, c_4, c_5, u_3, w_1, a_6, c_6, e_4, g_4, n_3, i_4, k_5, m_5, a'_1, h_4, f_4, d_4, z', w', v', m_3, t'_1, s'_1, r'_1, q'_1, p'_1, o', n'_1, m'_1, l'_1, k'_1, l_3, j'_2, k_3, j_3, b_3
  155. Polygon(H_1, M_2, A') → poly5, a'_2, h_5, m_6
  156. Polygon(Q_1, R_1, A, E', F', G', H', I', J') → poly6, q_6, r_6, a_7, e'_2, f'_2, g'_2, h'_2, i'_2, j'_3
  157. Translate(A, v) → A'_2
  158. Translate(R_1, v) → Z_2
  159. Translate(Q_1, v) → A_3
  160. Translate(J', v) → J''
  161. Translate(I', v) → I''
  162. Translate(H', v) → H''
  163. Translate(G', v) → G''
  164. Translate(F', v) → F''
  165. Translate(E', v) → E''
  166. Polygon(A'_2, Z_2, A_3, J'', I'', H'', G'', F'', E'') → poly2', a'_3, z_2, a_8, j'', i'', h'', g'', f'', e''
  167. Translate(R_2, v) → B_3
  168. Translate(S_2, v) → C_3
  169. Translate(T_2, v) → D_3
  170. Translate(U_2, v) → E_3
  171. Translate(V_2, v) → F_3
  172. Translate(W_2, v) → G_3
  173. Polygon(B_3, C_3, D_3, E_3, F_3, G_3) → poly3', b_5, c_7, d_5, e_5, f_5, g_5
  174. Translate(Q_1, v) → H_3
  175. Translate(P_1, v) → I_3
  176. Translate(O_1, v) → J_3
  177. Translate(N_1, v) → K_3
  178. Translate(M_1, v) → L_3
  179. Translate(L_1, v) → M_3
  180. Translate(K_1, v) → N_3
  181. Translate(J_1, v) → O_3
  182. Translate(B, v) → B'_4
  183. Translate(E, v) → E'_2
  184. Translate(F, v) → F'_2
  185. Translate(G, v) → G'_2
  186. Translate(H, v) → H'_2
  187. Translate(I, v) → I'_2
  188. Translate(J, v) → J'_2
  189. Translate(K, v) → K'_2
  190. Translate(L, v) → L'_2
  191. Translate(M, v) → M'_2
  192. Translate(N, v) → N'_2
  193. Translate(O, v) → O'_2
  194. Translate(P, v) → P'_2
  195. Translate(Q, v) → Q'_2
  196. Translate(R, v) → R'_2
  197. Translate(S, v) → S'_2
  198. Translate(T, v) → T'_2
  199. Translate(U, v) → U'_2
  200. Translate(V, v) → V'_2
  201. Translate(W, v) → W'_2
  202. Translate(Z, v) → Z'_2
  203. Translate(A_1, v) → P_3
  204. Translate(B_1, v) → Q_3
  205. Translate(C_1, v) → R_3
  206. Translate(C, v) → C'_2
  207. Translate(U_1, v) → S_3
  208. Translate(W_1, v) → T_3
  209. Translate(A_2, v) → U_3
  210. Translate(C_2, v) → V_3
  211. Translate(E_2, v) → W_3
  212. Translate(G_2, v) → Z_3
  213. Translate(I_2, v) → A_4
  214. Translate(K_2, v) → B_4
  215. Translate(M_2, v) → C_4
  216. Translate(A', v) → A''
  217. Translate(H_1, v) → D_4
  218. Translate(F_1, v) → E_4
  219. Translate(D_1, v) → F_4
  220. Translate(Z', v) → Z''
  221. Translate(W', v) → W''
  222. Translate(V', v) → V''
  223. Translate(T', v) → T''
  224. Translate(S', v) → S''
  225. Translate(R', v) → R''
  226. Translate(Q', v) → Q''
  227. Translate(P', v) → P''
  228. Translate(O', v) → O''
  229. Translate(N', v) → N''
  230. Translate(M', v) → M''
  231. Translate(L', v) → L''
  232. Translate(K', v) → K''
  233. Translate(J', v) → J''_1
  234. Polygon(H_3, I_3, F'_2, G'_2, H'_2, I'_2, J'_2, K'_2, L'_2, M'_2, N'_2, O'_2, J_3, P'_2, Q'_2, R'_2, S'_2, T'_2, U'_2, V'_2, W'_2, Z'_2, P_3, K_3, Q_3, R_3, C'_2, S_3, T_3, U_3, V_3, W_3, Z_3, A_4, L_3, B_4, C_4, A'', D_4, E_4, F_4, Z'', W'', V'', T'', M_3, S'', R'', Q'', P'', O'', N'', M'', L'', K'', J''_1, N_3, O_3, B'_4, E'_2) → poly4', h_6, e'_3, f'_3, g'_3, h'_3, i'_3, j'_4, k'_2, l'_2, m'_2, n'_2, i_5, o'_2, p'_2, q'_2, r'_2, s'_2, t'_2, u'_2, v'_2, w'_2, z'_2, j_5, p_5, q_7, r_7, c'_2, s_5, t_5, u_4, v_3, w_3, z_3, k_6, a_9, b_6, c_8, a'', d_6, e_6, f_6, z'', w'', v'', l_5, t'', s'', r'', q'', p'', o'', n'', m'', l'', k'', m_7, j''_1, n_5, o_3, b'_4
  235. Translate(H_1, v) → G_4
  236. Translate(M_2, v) → H_4
  237. Translate(A', v) → A''_1
  238. Polygon(G_4, H_4, A''_1) → poly5', a''_1, g_6, h_7
  239. Translate(Q_1, v) → I_4
  240. Translate(R_1, v) → J_4
  241. Translate(A, v) → A'_3
  242. Translate(E', v) → E''_1
  243. Translate(F', v) → F''_1
  244. Translate(G', v) → G''_1
  245. Translate(H', v) → H''_1
  246. Translate(I', v) → I''_1
  247. Translate(J', v) → J''_2
  248. Polygon(I_4, J_4, A'_3, E''_1, F''_1, G''_1, H''_1, I''_1, J''_2) → poly6', i_6, j_6, a'_4, e''_1, f''_1, g''_1, h''_1, i''_1, j''_2
  249. Translate(c_3, v) → p_6
  250. Translate(d_3, v) → s_6
  251. Translate(B, v) → B'_5
  252. Translate(E, v) → E'_3
  253. Segment(B'_5, E'_3) → e'_4
  254. Translate(E, v) → E'_4
  255. Translate(F, v) → F'_3
  256. Segment(E'_4, F'_3) → f'_4
  257. Translate(F, v) → F'_4
  258. Translate(G, v) → G'_3
  259. Segment(F'_4, G'_3) → g'_4
  260. Translate(G, v) → G'_4
  261. Translate(H, v) → H'_3
  262. Segment(G'_4, H'_3) → h'_4
  263. Translate(H, v) → H'_4
  264. Translate(I, v) → I'_3
  265. Segment(H'_4, I'_3) → i'_4
  266. Translate(I, v) → I'_4
  267. Translate(J, v) → J'_3
  268. Segment(I'_4, J'_3) → j'_5
  269. Translate(J, v) → J'_4
  270. Translate(K, v) → K'_3
  271. Segment(J'_4, K'_3) → k'_3
  272. Translate(K, v) → K'_4
  273. Translate(L, v) → L'_3
  274. Segment(K'_4, L'_3) → l'_3
  275. Translate(L, v) → L'_4
  276. Translate(M, v) → M'_3
  277. Segment(L'_4, M'_3) → m'_3
  278. Translate(M, v) → M'_4
  279. Translate(N, v) → N'_3
  280. Segment(M'_4, N'_3) → n'_3
  281. Translate(N, v) → N'_4
  282. Translate(O, v) → O'_3
  283. Segment(N'_4, O'_3) → p'_3
  284. Translate(O, v) → O'_4
  285. Translate(P, v) → P'_3
  286. Segment(O'_4, P'_3) → q'_3
  287. Translate(P, v) → P'_4
  288. Translate(Q, v) → Q'_3
  289. Segment(P'_4, Q'_3) → r'_3
  290. Translate(Q, v) → Q'_4
  291. Translate(R, v) → R'_3
  292. Segment(Q'_4, R'_3) → s'_3
  293. Translate(R, v) → R'_4
  294. Translate(S, v) → S'_3
  295. Segment(R'_4, S'_3) → t'_3
  296. Translate(S, v) → S'_4
  297. Translate(T, v) → T'_3
  298. Segment(S'_4, T'_3) → l_6
  299. Translate(T, v) → T'_4
  300. Translate(U, v) → U'_3
  301. Segment(T'_4, U'_3) → n_6
  302. Translate(U, v) → U'_4
  303. Translate(V, v) → V'_3
  304. Segment(U'_4, V'_3) → t_6
  305. Translate(V, v) → V'_4
  306. Translate(W, v) → W'_3
  307. Segment(V'_4, W'_3) → b_7
  308. Translate(W, v) → W'_4
  309. Translate(Z, v) → Z'_3
  310. Segment(W'_4, Z'_3) → d_7
  311. Translate(Z, v) → Z'_4
  312. Translate(A_1, v) → K_4
  313. Segment(Z'_4, K_4) → e_7
  314. Translate(A_1, v) → L_4
  315. Translate(B_1, v) → M_4
  316. Segment(L_4, M_4) → f_7
  317. Translate(B_1, v) → N_4
  318. Translate(C_1, v) → O_4
  319. Segment(N_4, O_4) → g_7
  320. Translate(C_1, v) → P_4
  321. Translate(C, v) → C'_3
  322. Segment(P_4, C'_3) → i_7
  323. Translate(B'_1, v) → Q_4
  324. Translate(E', v) → E''_2
  325. Segment(Q_4, E''_2) → e''_2
  326. Translate(E'_1, v) → R_4
  327. Translate(F', v) → F''_2
  328. Segment(R_4, F''_2) → f''_2
  329. Translate(F'_1, v) → S_4
  330. Translate(G', v) → G''_2
  331. Segment(S_4, G''_2) → g''_2
  332. Translate(G'_1, v) → T_4
  333. Translate(H', v) → H''_2
  334. Segment(T_4, H''_2) → h''_2
  335. Translate(H'_1, v) → U_4
  336. Translate(I', v) → I''_2
  337. Segment(U_4, I''_2) → i''_2
  338. Translate(I'_1, v) → V_4
  339. Translate(J', v) → J''_3
  340. Segment(V_4, J''_3) → j''_3
  341. Translate(J'_1, v) → W_4
  342. Translate(K', v) → K''_1
  343. Segment(W_4, K''_1) → k''_1
  344. Translate(K'_1, v) → Z_4
  345. Translate(L', v) → L''_1
  346. Segment(Z_4, L''_1) → l''_1
  347. Translate(L'_1, v) → A_5
  348. Translate(M', v) → M''_1
  349. Segment(A_5, M''_1) → m''_1
  350. Translate(M'_1, v) → B_5
  351. Translate(N', v) → N''_1
  352. Segment(B_5, N''_1) → n''_1
  353. Translate(N'_1, v) → C_5
  354. Translate(O', v) → O''_1
  355. Segment(C_5, O''_1) → p''_1
  356. Translate(O'_1, v) → D_5
  357. Translate(P', v) → P''_1
  358. Segment(D_5, P''_1) → q''_1
  359. Translate(P'_1, v) → E_5
  360. Translate(Q', v) → Q''_1
  361. Segment(E_5, Q''_1) → r''_1
  362. Translate(Q'_1, v) → F_5
  363. Translate(R', v) → R''_1
  364. Segment(F_5, R''_1) → s''_1
  365. Translate(R'_1, v) → G_5
  366. Translate(S', v) → S''_1
  367. Segment(G_5, S''_1) → t''_1
  368. Translate(S'_1, v) → H_5
  369. Translate(T', v) → T''_1
  370. Segment(H_5, T''_1) → j_7
  371. Translate(T'_1, v) → I_5
  372. Translate(U', v) → U''
  373. Segment(I_5, U'') → k_7
  374. Translate(U'_1, v) → J_5
  375. Translate(V', v) → V''_1
  376. Segment(J_5, V''_1) → l_7
  377. Translate(V'_1, v) → K_5
  378. Translate(W', v) → W''_1
  379. Segment(K_5, W''_1) → n_7
  380. Translate(W'_1, v) → L_5
  381. Translate(Z', v) → Z''_1
  382. Segment(L_5, Z''_1) → p_7
  383. Translate(Z'_1, v) → M_5
  384. Translate(D_1, v) → N_5
  385. Segment(M_5, N_5) → s_7
  386. Translate(E_1, v) → O_5
  387. Translate(F_1, v) → P_5
  388. Segment(O_5, P_5) → t_7
  389. Translate(G_1, v) → Q_5
  390. Translate(H_1, v) → R_5
  391. Segment(Q_5, R_5) → b_8
  392. Translate(I_1, v) → S_5
  393. Translate(C'_1, v) → T_5
  394. Segment(S_5, T_5) → d_8
  395. Translate(B, v) → B'_6
  396. Translate(J_1, v) → U_5
  397. Segment(B'_6, U_5) → e_8
  398. Translate(J_1, v) → V_5
  399. Translate(K_1, v) → W_5
  400. Segment(V_5, W_5) → f_8
  401. Translate(K_1, v) → Z_5
  402. Translate(L_1, v) → A_6
  403. Segment(Z_5, A_6) → g_8
  404. Translate(L_1, v) → B_6
  405. Translate(M_1, v) → C_6
  406. Segment(B_6, C_6) → h_8
  407. Translate(M_1, v) → D_6
  408. Translate(N_1, v) → E_6
  409. Segment(D_6, E_6) → i_8
  410. Translate(N_1, v) → F_6
  411. Translate(O_1, v) → G_6
  412. Segment(F_6, G_6) → j_8
  413. Translate(O_1, v) → H_6
  414. Translate(P_1, v) → I_6
  415. Segment(H_6, I_6) → k_8
  416. Translate(P_1, v) → J_6
  417. Translate(Q_1, v) → K_6
  418. Segment(J_6, K_6) → l_8
  419. Translate(Q_1, v) → L_6
  420. Translate(R_1, v) → M_6
  421. Segment(L_6, M_6) → m_8
  422. Translate(R_1, v) → N_6
  423. Translate(A, v) → A'_4
  424. Segment(N_6, A'_4) → n_8
  425. Translate(B'_3, v) → O_6
  426. Translate(U_1, v) → P_6
  427. Segment(O_6, P_6) → p_8
  428. Translate(V_1, v) → Q_6
  429. Translate(W_1, v) → R_6
  430. Segment(Q_6, R_6) → q_8
  431. Translate(Z_1, v) → S_6
  432. Translate(A_2, v) → T_6
  433. Segment(S_6, T_6) → r_8
  434. Translate(B_2, v) → U_6
  435. Translate(C_2, v) → V_6
  436. Segment(U_6, V_6) → s_8
  437. Translate(D_2, v) → W_6
  438. Translate(E_2, v) → Z_6
  439. Segment(W_6, Z_6) → t_8
  440. Translate(F_2, v) → A_7
  441. Translate(G_2, v) → B_7
  442. Segment(A_7, B_7) → b_9
  443. Translate(H_2, v) → C_7
  444. Translate(I_2, v) → D_7
  445. Segment(C_7, D_7) → c_9
  446. Translate(J_2, v) → E_7
  447. Translate(K_2, v) → F_7
  448. Segment(E_7, F_7) → d_9
  449. Translate(L_2, v) → G_7
  450. Translate(M_2, v) → H_7
  451. Segment(G_7, H_7) → e_9
  452. Translate(N_2, v) → I_7
  453. Translate(A'_1, v) → J_7
  454. Segment(I_7, J_7) → f_9
  455. Translate(Q_1, v) → K_7
  456. Translate(J', v) → J''_4
  457. Segment(K_7, J''_4) → g_9
  458. Translate(u, v) → u'_1
  459. Translate(B', v) → B''_1
  460. Translate(B'_2, v) → L_7
  461. Translate(B'', v) → B'''
  462. Translate(T_1, v) → M_7

AI Drawing Prompt Reference

Draw a square and from one vertex create a polyline made of many connected segments ending at the opposite vertex. Then translate the entire polyline by the vector along one side of the square to obtain a parallel copy, and connect corresponding vertices with segments to form a series of parallelograms. The construction illustrates how vector translation affects a polygonal region and allows computation of its area and definite integral.
The prompt above was compiled from the original drawing commands and is for reference only.