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四边形面积探索

四边形面积探索
几何画板及绘图内容加载中，请稍等……
2017-08-18 04:32
多边形角变换几何作图滑块动画
这个范例演示了如何通过几何作图来探索四边形的面积。核心数学概念包括多边形、角、变换和滑块动画。通过使用GeoGebra命令如Line、Polygon、Rotate和Mirror，可以动态地构造和观察四边形的变化。适合初中及以上学生和教师使用。

GeoGebra 命令及参数

  1. Line(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Point(b) → C
  4. Line(C, a) → c
  5. OrthogonalLine(B, a) → d
  6. Intersect(c, d) → D
  7. Segment(A, B) → e
  8. Point(e) → E
  9. Segment(E, C) → f
  10. Segment(E, D) → g
  11. Midpoint(f) → F
  12. Midpoint(g) → G
  13. Semicircle(E, D) → h
  14. OrthogonalLine(E, e) → i
  15. Intersect(c, i) → H
  16. Semicircle(B, H) → k
  17. Semicircle(D, E) → p_1
  18. Semicircle(H, A) → q
  19. Semicircle(C, E) → r
  20. Semicircle(E, C) → s
  21. Rotate(E, -α, G) → E'
  22. Angle(E', G, E) → β
  23. Rotate(B, -α, G) → B'
  24. Angle(B', G, B) → γ
  25. Rotate(D, -α, G) → D'
  26. Angle(D', G, D) → δ
  27. Polygon(E', B', D') → poly1, d', e', b'
  28. Rotate(A, α, F) → A'
  29. Angle(A, F, A') → ε
  30. Rotate(E, α, F) → E'_1
  31. Angle(E, F, E'_1) → ζ
  32. Rotate(C, α, F) → C'
  33. Angle(C, F, C') → η
  34. Polygon(A', C', E'_1) → poly2, e'_1, a', c'
  35. Midpoint(e) → I
  36. Midpoint(A, C) → J
  37. Midpoint(C, D) → K
  38. Midpoint(B, D) → L
  39. Polygon(J, I, L, K) → poly3, j_1, i_1, l, k_1
  40. Midpoint(j_1) → M
  41. Midpoint(i_1) → N
  42. Midpoint(l) → O
  43. Midpoint(k_1) → P
  44. Rotate(J, α, P) → J'
  45. Angle(J, P, J') → θ
  46. Rotate(C, α, P) → C'_1
  47. Angle(C, P, C'_1) → ι
  48. Rotate(K, α, P) → K'
  49. Angle(K, P, K') → κ
  50. Polygon(J', C'_1, K') → poly4, k', j', c'_1
  51. Rotate(K, -α, O) → K'_1
  52. Angle(K'_1, O, K) → λ
  53. Rotate(L, -α, O) → L'
  54. Angle(L', O, L) → μ
  55. Rotate(D, -α, O) → D'_1
  56. Angle(D'_1, O, D) → ν
  57. Polygon(K'_1, D'_1, L') → poly5, l', k'_1, d'_1
  58. Rotate(A, -α, M) → A'_1
  59. Angle(A'_1, M, A) → ξ
  60. Rotate(I, -α, M) → I'
  61. Angle(I', M, I) → ο
  62. Rotate(J, -α, M) → J'_1
  63. Angle(J'_1, M, J) → ρ
  64. Polygon(J'_1, A'_1, I') → poly6, i', j'_1, a'_1
  65. Rotate(B, α, N) → B'_1
  66. Angle(B, N, B'_1) → σ
  67. Rotate(L, α, N) → L'_1
  68. Angle(L, N, L'_1) → τ
  69. Rotate(I, α, N) → I'_1
  70. Angle(I, N, I'_1) → υ
  71. Polygon(B'_1, L'_1, I'_1) → poly7, i'_1, b'_1, l'_1
  72. Segment(A', C') → n_2
  73. Polygon(C, E, D) → poly8, d_1, c_1, e_1
  74. Midpoint(A, C) → Q
  75. Midpoint(D, C) → R
  76. Midpoint(A, B) → S
  77. Midpoint(B, D) → T
  78. Segment(Q, T) → t
  79. Segment(S, R) → a_1
  80. Intersect(t, a_1) → U
  81. Point(e) → V
  82. Mirror(V, U) → V'
  83. Polygon(C, Q, V, B, T, V') → poly9, c_2, q_1, v, b_1, t_1, v'
  84. Rotate(A, α, Q) → A'_2
  85. Angle(A, Q, A'_2) → ϕ
  86. Rotate(V, α, Q) → V'_1
  87. Angle(V, Q, V'_1) → χ
  88. Polygon(Q, V'_1, A'_2) → poly10, a'_2, q_2, v'_1
  89. Rotate(V', α, T) → V''
  90. Angle(V', T, V'') → ψ
  91. Rotate(D, α, T) → D'_2
  92. Angle(D, T, D'_2) → ω
  93. Polygon(V'', D'_2, T) → poly11, t_2, v'', d'_2
  94. Segment(C, D) → f_1
  95. Segment(C, V') → g_1
  96. Segment(V, B) → h_1
  97. Segment(Q, C) → l_1
  98. Midpoint(A, C) → W
  99. Midpoint(B, D) → Z
  100. Segment(V, B) → m_1
  101. Point(m_1) → A_1
  102. Polygon(W, C, D, Z, A_1, V) → poly12, w, c_3, d_2, z, u_2, v_1
  103. Rotate(A_1, -α, Z) → B_1
  104. Angle(B_1, Z, A_1) → α_1
  105. Rotate(B, -α, Z) → B'_2
  106. Angle(B'_2, Z, B) → β_1
  107. Polygon(B_1, Z, B'_2) → poly13, b'_2, b_2, z_1
  108. Segment(E, C) → n_1
  109. Segment(C, D) → r_1
  110. Segment(E, D) → s_1
  111. Midpoint(s_1) → C_1
  112. Rotate(E, α, C_1) → E'_2
  113. Angle(E, C_1, E'_2) → γ_1
  114. Rotate(C, α, C_1) → C'_2
  115. Angle(C, C_1, C'_2) → δ_1
  116. Rotate(D, α, C_1) → D'_3
  117. Angle(D, C_1, D'_3) → ε_1
  118. Polygon(E'_2, C'_2, D'_3) → poly14, d'_3, e'_2, c'_2
  119. Midpoint(A, C) → D_1
  120. Midpoint(B, D) → E_1
  121. Mirror(V, D_1) → V'_2
  122. Mirror(A_1, E_1) → F_1
  123. Polygon(V, V'_2, F_1, A_1) → poly15, v_2, v'_2, f_2, a_2
  124. Midpoint(A_1, F_1) → G_1
  125. Rotate(A_1, α, G_1) → H_1
  126. Angle(A_1, G_1, H_1) → ζ_1
  127. Rotate(V, α, G_1) → V'_3
  128. Angle(V, G_1, V'_3) → η_1
  129. Rotate(V'_2, α, G_1) → I_1
  130. Angle(V'_2, G_1, I_1) → θ_1
  131. Rotate(F_1, α, G_1) → J_1
  132. Angle(F_1, G_1, J_1) → ι_1
  133. Polygon(V'_3, I_1, J_1, H_1) → poly16, v'_3, i_2, j_2, h_2
  134. Polygon(E'_1, A', C') → poly17, c'_3, e'_3, a'_3
  135. Polygon(A, C, D, B) → poly18, a_3, c_4, d_3, b_3
  136. Polygon(C, E, D) → poly19, d_4, c_5, e_2
  137. Polygon(E', B', D') → poly20, d'_4, e'_4, b'_3
  138. Polygon(A'_1, J'_1, I') → poly21, i'_2, a'_4, j'_2
  139. Polygon(B'_1, I'_1, L'_1) → poly22, l'_2, b'_4, i'_3
  140. Polygon(J', K', C'_1) → poly23, c'_4, j'_3, k'_2
  141. Polygon(K'_1, D'_1, L') → poly24, l'_3, k'_3, d'_5
  142. Polygon(A, C, D, B) → poly25, a_4, c_6, d_5, b_4
  143. Polygon(I, J, K, L) → poly26, i_3, j_3, k_2, l_2
  144. Polygon(V'_1, A'_2, Q) → poly27, q_3, v'_4, a'_5
  145. Polygon(D'_2, T, V'') → poly28, v''_1, d'_6, t_3
  146. Polygon(V, V'_1, V', V'') → poly29, v_3, v'_5, v'_6, v''_2
  147. Polygon(A, C, D, B) → poly30, a_5, c_7, d_6, b_5
  148. Point(b_5) → K_1
  149. Polygon(V'_1, A'_2, Q) → poly31, q_4, v'_7, a'_6
  150. Polygon(B_1, Z, B'_2) → poly32, b'_5, b_6, z_2
  151. Polygon(V, V'_1, B_1, A_1) → poly33, v_4, v'_8, b_7, a_6
  152. Polygon(A, C, D, B) → poly34, a_7, c_8, d_7, b_8
  153. Polygon(E, C, D) → poly35, d_8, e_3, c_9
  154. Polygon(D'_3, E'_2, C'_2) → poly36, c'_5, d'_7, e'_5
  155. Polygon(C'_2, E, C, D) → poly37, c'_6, e_4, c_{10}, d_9
  156. Polygon(V, V'_2, F_1, A_1) → poly38, v_5, v'_9, f_3, a_8
  157. Polygon(J_1, H_1, V'_3, I_1) → poly39, j_4, h_3, v'_{10}, i_4
  158. Polygon(V, V'_2, V'_3, I_1) → poly40, v_6, v'_{11}, v'_{12}, i_5
  159. Midpoint(A, E) → L_1
  160. Midpoint(E, B) → M_1
  161. Mirror(C, L_1) → C'_3
  162. Polygon(A, C, D, B) → poly41, a_9, c_{11}, d_{10}, b_9
  163. Polygon(L_1, C, D, M_1) → poly42, l_3, c_{12}, d_{11}, m_2
  164. Mirror(A, L_1) → A'_3
  165. Mirror(A'_3, L_1) → O_1
  166. Intersect(l_3, c_{12}) → P_1
  167. Rotate(O_1, -α, L_1) → Q_1
  168. Angle(Q_1, L_1, O_1) → κ_1
  169. Rotate(P_1, -α, L_1) → R_1
  170. Angle(R_1, L_1, P_1) → λ_1
  171. Polygon(Q_1, R_1, L_1) → poly43, l_4, q_5, r_2
  172. Rotate(D, α, M_1) → D'_4
  173. Angle(D, M_1, D'_4) → μ_1
  174. Rotate(B, α, M_1) → B'_3
  175. Angle(B, M_1, B'_3) → ν_1
  176. Polygon(M_1, B'_3, D'_4) → poly44, d'_8, m_3, b'_6
  177. Polygon(R_1, C, D) → poly45, d_{12}, r_3, c_{13}
  178. Circle(C, D) → p_2
  179. Line(A, B) → g_2
  180. Intersect(p_2, g_2) → N_1, S_1
  181. Polygon(A, C, D, B) → poly46, a_{10}, c_{14}, d_{13}, b_{10}
  182. Polygon(C, S_1, B, D) → poly47, c_{15}, s_2, b_{11}, d_{14}
  183. Segment(A, B) → g_3
  184. Midpoint(g_3) → T_1
  185. Rotate(A, -α, T_1) → A'_4
  186. Angle(A'_4, T_1, A) → ξ_1
  187. Segment(C, D) → k_3
  188. Midpoint(c_{14}) → U_1
  189. Rotate(C, -α, U_1) → C'_4
  190. Angle(C'_4, U_1, C) → ο_1
  191. Midpoint(S_1, B) → V_1
  192. Mirror(A, V_1) → A'_5
  193. Midpoint(S_1, A'_5) → W_1
  194. Rotate(S_1, -α, W_1) → Z_1
  195. Angle(Z_1, W_1, S_1) → ρ_1
  196. Polygon(A'_4, C'_4, Z_1) → poly48, z_3, a'_7, c'_7
  197. Polygon(C, S_1, Z_1, D) → poly49, c_{16}, s_3, z_4, d_{15}

AI 绘图提示语参考

绘制一个四边形，通过构造线段、点和圆弧来探索四边形的面积。首先画出线段AB，然后构造AB的垂线，找到垂足C，接着画出线段BC，构造其垂线，找到垂足D。连接AD和BC，找到AD和BC的中点F和G，构造以E、D和E、C为直径的圆弧，旋转点E、B、D得到E'、B'、D'，构造四边形E'B'D'。接着旋转点A、E、C得到A'、E'_1、C'，构造四边形A'C'E'_1。最后找到线段AB、EC的中点I、J、L、K，构造四边形JKIL，旋转点J、C、K得到J'、C'_1、K'。
以上提示语为根据原始绘图命令整理而来，仅供参考。