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Parallel and Square Animation Demo

Parallel and Square Animation Demo
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2016-06-29 09:05
PolygonVectorCoordinate GeometryTransformationGeometric Construction
In collections CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
This example demonstrates the geometric properties and transformation relationships between parallelograms and squares through animation. It involves core mathematical concepts such as vectors, geometric transformations, coordinate geometry, and geometric construction. The construction utilizes commands and objects like Segment, Vector, Line, Slope, PointIn, Circle, Dilate, Translate, OrthogonalLine, Intersect, Rotate, Angle, Min, CircleSector, Polygon, Midpoint, Point in GeoGebra. Suitable for high school mathematics teaching to understand geometric transformations and properties, enhancing spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Segment(A, B) → h
  2. Vector(A, B) → u
  3. Line(A, B) → a
  4. Slope(a) → m
  5. PointIn(c) → C
  6. Circle(C, h) → p
  7. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], C) → A'
  8. Vector(A, A') → v
  9. Translate(B, v) → B'
  10. Translate(A, v) → A'_1
  11. Segment(A'_1, B') → h'
  12. Segment(A, C) → d
  13. OrthogonalLine(C, d) → g_1
  14. OrthogonalLine(A, d) → f_1
  15. Circle(A, d) → t
  16. Intersect(t, f_1, 2) → H
  17. OrthogonalLine(H, g_1) → h_1
  18. Intersect(g_1, h_1) → I
  19. Translate(C, u) → C'
  20. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], C) → C''_1
  21. Vector(C', C''_1) → v_1
  22. Rotate(B', If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C') → B''
  23. Rotate(A'_1, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C') → N
  24. Segment(N, B'') → h''
  25. Segment(C, C') → n
  26. OrthogonalLine(C, n) → q
  27. Intersect(p, q, 2) → F
  28. Angle(F, C, I) → γ
  29. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], C') → C'_1
  30. Vector(C, C'_1) → w
  31. Translate(C, w) → C'_2
  32. Translate(A, w) → A'_2
  33. Segment(A'_2, C'_2) → d'
  34. Min(h', d') → o
  35. Circle(C', (0.15 * o)) → e_1
  36. Intersect(e_1, h', 1) → P
  37. Intersect(e_1, h'', 1) → O
  38. CircleSector(C', O, P) → s_1
  39. Translate(s_1, v_1) → r_2
  40. Line(C, a) → e
  41. Polygon(A, B, C') → poly1, c', a_1, b_1
  42. Polygon(A, C', C) → poly2, c_1, a_2, c'_1
  43. Polygon(A, B, C', C) → poly3, a_3, b_2, c'_2, c_2
  44. Circle(B, d) → f
  45. Segment(B, C') → g
  46. OrthogonalLine(B, g) → i
  47. OrthogonalLine(C', g) → j
  48. Intersect(f, i, 1) → D
  49. OrthogonalLine(D, i) → l
  50. Intersect(j, l) → E
  51. OrthogonalLine(C', n) → r
  52. OrthogonalLine(F, q) → s
  53. Intersect(r, s) → G
  54. Circle(A, h) → d_1
  55. OrthogonalLine(A, h) → i_1
  56. Intersect(d_1, i_1, 1) → J
  57. OrthogonalLine(J, i_1) → j_1
  58. OrthogonalLine(B, h) → k_1
  59. Intersect(j_1, k_1) → K
  60. Rotate(A'_2, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], B) → L
  61. Rotate(C'_2, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], B) → M
  62. Segment(L, M) → d''
  63. Circle(B, (0.25 * o)) → p_1
  64. Circle(C, (0.25 * o)) → q_1
  65. Circle(A, (0.25 * o)) → r_1
  66. Rotate(A, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C) → A'_3
  67. Rotate(C, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C) → C'_3
  68. Segment(A'_3, C'_3) → d'_1
  69. Rotate(A, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], A) → A'_4
  70. Rotate(B, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], A) → B'_1
  71. Segment(A'_4, B'_1) → h'_1
  72. Rotate(N, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], G) → N'
  73. Rotate(B'', If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], G) → B'''
  74. Segment(N', B''') → h'''
  75. Rotate(A'_3, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], I) → Q
  76. Rotate(C'_3, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], I) → R
  77. Segment(Q, R) → l_1
  78. Rotate(A'_4, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], J) → S
  79. Rotate(B'_1, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], J) → T
  80. Segment(S, T) → m_1
  81. Rotate(L, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], D) → L'
  82. Rotate(M, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], D) → M'
  83. Segment(L', M') → d'''
  84. Rotate(N', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], F) → N''
  85. Rotate(B''', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], F) → U
  86. Segment(N'', U) → n_1
  87. Rotate(Q, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], H) → Q'
  88. Rotate(R, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], H) → R'
  89. Segment(Q', R') → t_1
  90. Rotate(S, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], K) → S'
  91. Rotate(T, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], K) → T'
  92. Segment(S', T') → f_2
  93. Rotate(L', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], E) → L''
  94. Rotate(M', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], E) → M''
  95. Segment(L'', M'') → g_2
  96. Polygon(C, C', G, F) → poly4, c_3, c'_3, g_3, f_3
  97. Polygon(A, B, K, J) → poly5, a_4, b_3, k_2, j_2
  98. Polygon(B, D, E, C') → poly6, b_4, d_2, e_2, c'_4
  99. Polygon(A, C, I, H) → poly7, a_5, c_4, i_2, h_2
  100. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], G) → C''
  101. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], A) → C''_2
  102. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], B) → C''_3
  103. Vector(C', C'') → u_1
  104. Vector(C', C''_2) → w_1
  105. Vector(C', C''_3) → u_2
  106. Translate(s_1, u_1) → p_2
  107. Translate(s_1, w_1) → q_2
  108. Translate(s_1, u_2) → s_2
  109. Line(C, C') → l_2
  110. Intersect(l_2, l_1) → V
  111. Rotate(p_2, If[6 < k ≤ 7, ((k - 6) * π / 2), π / 2], G) → d_3
  112. Rotate(q_2, If[6 < k ≤ 7, ((k - 6) * π), π], A) → e_3
  113. Line(B, u_1) → m_2
  114. Intersect(l_2, m_2) → W
  115. Rotate(s_2, If[6 < k ≤ 7, ((k - 6) * (π + γ)), π + γ], B) → h_3
  116. Dilate(G, If[7 < k ≤ 8, 8 - k, 0], F) → G'
  117. Dilate(C, If[7 < k ≤ 8, 8 - k, 0], I) → C'_4
  118. Dilate(A, If[7 < k ≤ 8, 8 - k, 0], J) → A'_5
  119. Dilate(B, If[7 < k ≤ 8, 8 - k, 0], D) → B'_2
  120. Vector(G, G') → v_2
  121. Vector(C, C'_4) → w_2
  122. Vector(A, A'_5) → n_2
  123. Vector(B, B'_2) → u_3
  124. Translate(d_3, v_2) → k_3
  125. Rotate(r_2, If[6 < k ≤ 7, ((k - 6) * γ), γ], C) → t_2
  126. Translate(t_2, w_2) → p_3
  127. Translate(e_3, n_2) → q_3
  128. Translate(h_3, u_3) → r_3
  129. Rotate(k_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], F) → s_3
  130. Rotate(p_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], I) → t_3
  131. Rotate(r_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], D) → d_4
  132. Rotate(q_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], J) → e_4
  133. Dilate(F, If[9 < k ≤ 10, 10 - k, 0], C) → F'
  134. Dilate(I, If[9 < k ≤ 10, 10 - k, 0], H) → I'
  135. Dilate(J, If[9 < k ≤ 10, 10 - k, 0], K) → J'
  136. Dilate(D, If[9 < k ≤ 10, 10 - k, 0], E) → D'
  137. Vector(D, D') → v_3
  138. Vector(J, J') → w_3
  139. Vector(I, I') → i_3
  140. Vector(F, F') → j_3
  141. Translate(t_3, i_3) → f_4
  142. Translate(s_3, j_3) → g_4
  143. Translate(d_4, v_3) → h_4
  144. Translate(e_4, w_3) → k_4
  145. Rotate(h_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], D') → p_4
  146. Rotate(f_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], H) → q_4
  147. Rotate(g_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], C) → r_4
  148. Rotate(k_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], K) → s_4
  149. Angle(C, C', B) → α
  150. Rotate(s_1, If[11 < k ≤ 12, ((k - 11) * (π / 2 + α)), π / 2 + α], C') → t_4
  151. Rotate(e_3, If[11 < k ≤ 12, ((k - 11) * (α + π / 2)), α + π / 2], A) → c_5
  152. Rotate(h_3, If[11 < k ≤ 12, ((k - 11) * ((3 * π / 2) - α)), (3 * π / 2) - α], B) → d_5
  153. Segment(B, E) → m_3
  154. Segment(A, K) → n_3
  155. Segment(C, H) → i_4
  156. Midpoint(i_4) → B_1
  157. Midpoint(n_3) → C_1
  158. Midpoint(m_3) → D_1
  159. Segment(C', F) → l_3
  160. Midpoint(l_3) → Z
  161. Dilate(C, If[12 < k ≤ 13, 13 - k, 0], Z) → C''_4
  162. Dilate(A, If[12 < k ≤ 13, 13 - k, 0], B_1) → C'_5
  163. Dilate(B, If[12 < k ≤ 13, 13 - k, 0], C_1) → A'_6
  164. Dilate(C', If[12 < k ≤ 13, 13 - k, 0], D_1) → B'_3
  165. Point(h'_1) → E_1
  166. Segment(B, A'_6) → m_4
  167. Segment(C', B'_3) → n_4
  168. Segment(C, C''_4) → j_4
  169. Dilate(Z, If[13 < k ≤ 14, 14 - k, 0], G) → Z'
  170. Dilate(B_1, If[13 < k ≤ 14, 14 - k, 0], I) → A_1
  171. Dilate(C_1, If[13 < k ≤ 14, 14 - k, 0], J) → F_1
  172. Dilate(D_1, If[13 < k ≤ 14, 14 - k, 0], D) → G_1
  173. Segment(Z, Z') → f_5
  174. Segment(D_1, G_1) → g_5
  175. Segment(B_1, A_1) → h_5
  176. Segment(C_1, F_1) → i_5
  177. Rotate(C, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], Z) → C'_6
  178. Rotate(C''_4, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], Z) → H_1
  179. Segment(C'_6, H_1) → j_5
  180. Rotate(A, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], B_1) → I_1
  181. Rotate(B_1, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], B_1) → J_1
  182. Segment(I_1, J_1) → k_5
  183. Rotate(B, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], C_1) → B'_4
  184. Rotate(A'_6, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], C_1) → K_1
  185. Segment(B'_4, K_1) → l_5
  186. Rotate(C', If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], D_1) → C''_5
  187. Rotate(B'_3, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], D_1) → L_1
  188. Segment(C''_5, L_1) → m_5
  189. Dilate(D_1, If[15 < k ≤ 16, 16 - k, 0], Z) → M_1
  190. Segment(D_1, M_1) → n_5
  191. Rotate(D_1, If[16 < k ≤ 17, ((-(k - 16)) * π), (-π)], Z) → N_1
  192. Rotate(M_1, If[16 < k ≤ 17, ((-(k - 16)) * π), (-π)], Z) → O_1
  193. Rotate(D_1, If[16 < k ≤ 17, ((-(k - 16)) * π / 2), ((-π)) / 2], Z) → P_1
  194. Rotate(M_1, If[16 < k ≤ 17, ((-(k - 16)) * π / 2), ((-π)) / 2], Z) → Q_1
  195. Segment(P_1, Q_1) → p_5
  196. Rotate(P_1, If[17 < k ≤ 18, ((-(k - 17)) * π / 2), ((-π)) / 2], B_1) → R_1
  197. Rotate(Q_1, If[17 < k ≤ 18, ((-(k - 17)) * π / 2), ((-π)) / 2], B_1) → S_1
  198. Segment(R_1, S_1) → q_5
  199. Rotate(R_1, If[18 < k ≤ 19, ((-(k - 18)) * π / 2), ((-π)) / 2], C_1) → T_1
  200. Rotate(S_1, If[18 < k ≤ 19, ((-(k - 18)) * π / 2), ((-π)) / 2], C_1) → U_1
  201. Segment(T_1, U_1) → r_5
  202. Polygon(D_1, Z, B_1, C_1) → poly8, d_6, z_1, b_5, c_6
  203. Dilate(E, If[19 < k ≤ 20, 20 - k, 0], D_1) → E'
  204. Dilate(F, If[19 < k ≤ 20, 20 - k, 0], Z) → F'_1
  205. Dilate(H, If[19 < k ≤ 20, 20 - k, 0], B_1) → H'
  206. Dilate(K, If[19 < k ≤ 20, 20 - k, 0], C_1) → K'
  207. Vector(K, K') → u_4
  208. Vector(H, H') → v_4
  209. Vector(E, E') → w_4
  210. Vector(F, F'_1) → u_5
  211. Translate(s_3, u_5) → e_5
  212. Translate(q_4, v_4) → s_5
  213. Translate(s_4, u_4) → t_5
  214. Translate(p_4, w_4) → e_6
  215. Line(D_1, g_2) → f_6
  216. Point(f_6) → W_1
  217. Angle(Z, D_1, W_1) → β
  218. Rotate(e_6, If[20 < k ≤ 21, ((-(k - 20)) * β), (-β)], D_1) → g_6
  219. Rotate(s_5, If[20 < k ≤ 21, ((-(k - 20)) * β), (-β)], B_1) → h_6
  220. Line(C_1, f_2) → i_6
  221. Point(i_6) → Z_1
  222. Angle(D_1, C_1, Z_1) → δ
  223. Rotate(t_5, If[20 < k ≤ 21, ((-(k - 20)) * δ), (-δ)], C_1) → k_6
  224. Rotate(e_5, If[20 < k ≤ 21, ((-(k - 20)) * δ), (-δ)], Z) → p_6
  225. Segment(A, C'_5) → l_4