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梯形面积交互式探索

梯形面积交互式探索
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2018-11-13 10:03
三角函数解析几何几何作图
所属专题 Perimeter & Area
本范例演示如何通过交互式操作探索梯形面积的计算方法。涉及的核心数学概念包括梯形面积公式、中位线性质、逻辑推理以及勾股定理和三角函数的应用。作图要点包括使用 Ray, Point, Segment, Slope, PointIn, Midpoint, Line, Dilate, Rotate, Polygon, OrthogonalLine, Intersect, Min, Circle, Angle, CircleSector, Vector, Translate 等GeoGebra命令和对象。适合初中及以上学生用于理解梯形面积计算和几何作图。

GeoGebra 命令及参数

  1. Ray(A, B) → g
  2. Point(g) → C
  3. Segment(A, C) → i
  4. Slope(i) → m
  5. PointIn(a) → D
  6. Midpoint(D, A) → S
  7. Line(A, B) → f
  8. Ray(A, C) → h
  9. Line(D, i) → j
  10. Point(j) → E
  11. Ray(D, E) → k
  12. Point(k) → F
  13. Segment(D, F) → l
  14. Segment(A, D) → n
  15. Segment(F, C) → p
  16. Midpoint(p) → G
  17. Dilate(A, If[0 < c ≤ 1, 1 - c, 0], C) → A'
  18. Segment(A, A') → q
  19. Dilate(C, If[1 < c ≤ 2, 2 - c, 0], G) → C'
  20. Dilate(G, If[2 < c ≤ 3, 3 - c, 0], F) → G'
  21. Segment(G, G') → s
  22. Rotate(C, If[3 < c ≤ 4, ((c - 3) * pi), pi], G) → C'_1
  23. Rotate(C', If[3 < c ≤ 4, ((c - 3) * pi), pi], G) → C''
  24. Dilate(F, If[4 < c ≤ 5, 5 - c, 0], D) → F'
  25. Segment(F, F') → t
  26. Dilate(D, If[5 < c ≤ 6, 6 - c, 0], S) → D'
  27. Dilate(D, If[6.5 < c ≤ 7.5, 7.5 - c, 0], G) → D'_1
  28. Polygon(A, C, G, S) → poly1, e, f_1, g_1, k_1
  29. Rotate(G, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → G'_1
  30. Rotate(F, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → F'_1
  31. Rotate(D, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → D'_2
  32. Rotate(D, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → D'_3
  33. Rotate(D'_1, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → H
  34. Rotate(C'_1, If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → I
  35. Rotate(C'', If[8 < c ≤ 9, ((-(c - 8)) * pi), pi], G) → C'''
  36. OrthogonalLine(D, q) → i_1
  37. Intersect(f, q) → J
  38. Dilate(D, If[6 < c ≤ 7, 7 - c, 0], J) → D'_4
  39. Segment(D, D'_4) → j_1
  40. Min(t, q) → o
  41. Circle(J, (0.15 * o)) → e_1
  42. Intersect(e_1, i_1) → K, L
  43. Intersect(e_1, f) → M, N
  44. Angle(D, J, A) → α
  45. Angle(C, A, D) → β
  46. Intersect(i_1, f) → O
  47. Dilate(D, If[6 < c ≤ 7, 7 - c, 0], O) → D'_5
  48. Point(f) → P
  49. Point(f) → Q
  50. Circle(C, t) → p_1
  51. Intersect(p_1, f, 2) → R
  52. Ray(R, Q) → m_1
  53. Angle(F, C, A) → δ
  54. Line(A, D) → l_1
  55. Rotate(D, If[6 < c ≤ 7, ((-(c - 6)) * pi), (-pi)], S) → D'_6
  56. Rotate(D', If[6 < c ≤ 7, ((-(c - 6)) * pi), (-pi)], S) → D''
  57. Segment(F, C) → r
  58. Segment(D, A) → d
  59. Midpoint(D, O) → T
  60. Polygon(S, G, F, D) → q1, s_1, g_2, f_2, d_1
  61. Circle(A, (0.4 * Distance[A, S])) → c_1
  62. Intersect(c_1, d, 1) → U
  63. Intersect(c_1, q, 1) → V
  64. Dilate(D, If[5 < c ≤ 6, 6 - c, 0], T) → D'_7
  65. Segment(D, D'_7) → n_1
  66. Dilate(T, If[6 < c ≤ 7, 7 - c, 0], O) → T'
  67. Segment(T, T') → q_1
  68. Angle(C, O, D) → γ
  69. Rotate(D, If[7 < c ≤ 8 - ((c - 7) * pi), (-pi)], T) → D'_8
  70. Rotate(D'_7, If[7 < c ≤ 8 - ((c - 7) * pi), (-pi)], T) → W
  71. Rotate(D, If[7 < c ≤ 8, ((-(c - 7)) * pi), (-pi)], T) → D'_9
  72. Rotate(D'_7, If[7 < c ≤ 8, ((-(c - 7)) * pi), (-pi)], T) → Z
  73. Segment(D'_9, Z) → r_1
  74. Dilate(T, If[8 < c ≤ 9, 9 - c, 0], S) → T'_1
  75. Dilate(T, If[8 < c ≤ 9, 9 - c, 0], G) → T'_2
  76. Segment(T, T'_2) → t_1
  77. Segment(T, T'_1) → a_1
  78. Angle(V, A, U) → ε
  79. Rotate(V, If[9 < c ≤ 10, ((c - 9) * ε), ε], A) → V'
  80. CircleSector(A, V, V') → h_1
  81. Dilate(A, If[10.5 < c ≤ 11.5, 11.5 - c, 0], S) → A'_1
  82. Vector(A, A'_1) → u
  83. Translate(h_1, u) → c_2
  84. Rotate(S, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → S'
  85. Rotate(G, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → G'_2
  86. Rotate(F, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → F'_2
  87. Rotate(D, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → D'_{10}
  88. Polygon(S', G'_2, F'_2, D'_{10}) → q1', s', g'_2, f'_2, d'_{10}
  89. Rotate(F, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → F'_3
  90. Rotate(F', If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → F''
  91. Segment(F'_3, F'') → t'
  92. Segment(G, F'_2) → b_1
  93. Rotate(S, If[12 < c ≤ 13, ((-(c - 12)) * pi), pi], G) → S'_1
  94. Segment(S'_1, D'_{10}) → h_2
  95. Segment(S'_1, G) → i_2
  96. Segment(F, G) → j_2
  97. Segment(G, C) → k_2
  98. Dilate(A, If[13 < c ≤ 14, 14 - c, 0], D'_{10}) → A'_2
  99. Vector(A, A'_2) → v
  100. Translate(h_1, v) → d_2