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斜率发现：非垂直直线的动态作图

斜率发现：非垂直直线的动态作图
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2019-07-15 23:47
向量解析几何几何作图
本范例演示如何通过动态几何工具绘制非垂直直线的方程。核心数学概念包括斜率、向量变换与直线方程的几何意义。作图要点涉及使用GeoGebra命令如Vector、Translate、Dilate、Polygon、Segment、Line等构建图形，并利用Midpoint、Slope等工具验证斜率计算。适合高中生学习线性方程与几何变换，或作为数学建模的实践案例。

GeoGebra 命令及参数

  1. Vector((0, 0), (1, 0)) → u
  2. Translate(A, u) → A'
  3. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], A') → A'_1
  4. Dilate(A'_1, If[1 < o ≤ 2, 2 - o, 0], B) → H
  5. Dilate((0, 0), If[-1 < o ≤ 0, (-o), 0], A) → D_1
  6. Translate(G, u) → G'
  7. Translate(B, u) → B'
  8. Translate(C, u) → C'
  9. Translate(D, u) → D'
  10. Polygon(A, A', B) → poly1, b, a_1, a'
  11. Polygon(B, B', C) → poly2, c, b_1, b'
  12. Polygon(C, C', D) → poly3, d, c_1, c'
  13. Polygon(D, D', E) → poly4, e, d_1, d'
  14. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], B') → B'_1
  15. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], C') → C'_1
  16. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], D') → D'_1
  17. Segment(A, A'_1) → p
  18. Segment(B, B'_1) → q
  19. Segment(C, C'_1) → r
  20. Segment(D, D'_1) → s
  21. Dilate(A'_1, If[2 < o ≤ 3, 3 - o, 0], B'_1) → I
  22. Vector(A'_1, I) → v
  23. Translate(A'_1, v) → J
  24. Translate(H, v) → H'
  25. Dilate(B'_1, If[3 < o ≤ 4, 4 - o, 0], C'_1) → K
  26. Vector(B'_1, K) → w
  27. Translate(J, w) → J'
  28. Translate(H', w) → H''
  29. Dilate(C'_1, If[4 < o ≤ 5, 5 - o, 0], D'_1) → M
  30. Vector(C'_1, M) → u_1
  31. Translate(J', u_1) → J''
  32. Translate(H'', u_1) → H'''
  33. Translate(L, u) → L'
  34. Translate(F, u) → F'
  35. Line(A, B) → f_1
  36. IntegralBetween(f, g, 0, 1) → e_1
  37. Translate(O, u) → O'
  38. IntegralBetween(f, g_1, 1, 2) → h_1
  39. IntegralBetween(f, h, 2, 3) → i_1
  40. Dilate(O, If[0 < p_1 ≤ 1, 1 - p_1, 0], O') → O'_1
  41. Dilate(L, If[0 < p_1 ≤ 1, 1 - p_1, 0], L') → L'_1
  42. Dilate(G, If[0 < p_1 ≤ 1, 1 - p_1, 0], G') → G'_1
  43. Segment(G, G'_1) → k_1
  44. Segment(O, O'_1) → l_1
  45. Segment(L, L'_1) → m_1
  46. Dilate(G'_1, If[1 < p_1 ≤ 2, 2 - p_1, 0], O) → P
  47. Segment(G'_1, P) → q_1
  48. Midpoint(O'_1, L) → Q
  49. Midpoint(L'_1, F) → R
  50. Dilate(G'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], O'_1) → S
  51. Vector(G'_1, S) → v_1
  52. Translate(G'_1, v_1) → T
  53. Translate(P, v_1) → P'
  54. Segment(T, P') → j_1
  55. Dilate(G'_1, If[3 < p_1 ≤ 4, 4 - p_1, 0], Q) → U
  56. Vector(G'_1, U) → w_1
  57. Translate(G'_1, w_1) → V
  58. Translate(P, w_1) → P'_1
  59. Segment(V, P'_1) → n_1
  60. Segment(O'_1, L) → r_1
  61. Dilate(O'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], L'_1) → W
  62. Vector(O'_1, W) → s_1
  63. Translate(O'_1, s_1) → Z
  64. Translate(L, s_1) → L'_2
  65. Segment(Z, L'_2) → t_1
  66. Dilate(O'_1, If[5 < p_1 ≤ 6, 6 - p_1, 0], R) → A_1
  67. Vector(O'_1, A_1) → u_2
  68. Translate(O'_1, u_2) → B_1
  69. Translate(L, u_2) → L'_3
  70. Segment(B_1, L'_3) → f_2
  71. Segment(C_1, A) → g_2
  72. Slope(f_1) → a_2
  73. Textfield(m) → InputBox1
  74. Distance(C_1, A) → distanceC1A
  75. Distance(A'_1, H) → c_2
  76. Vector(E_1, D_1) → v_2
  77. Vector(A'_1, H) → t
  78. Distance(E_1, D_1) → distanceE1D1
  79. Vector(J, H') → w_2
  80. Vector(J', H'') → c_2
  81. Vector(J'', H''') → d_2
  82. Distance(J, C) → distanceJC
  83. Distance(J', D) → distanceJ'D
  84. Distance(J'', E) → distanceJ''E