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Probability and Integral Calculation of a Ball Passing Through a Triangle

Probability and Integral Calculation of a Ball Passing Through a Triangle
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2019-07-05 15:08
VectorCalculusProbability & StatisticsOptimizationGeometric Construction
This example demonstrates the process of a ball passing through a triangular region and calculates its probability, involving core mathematical concepts such as Probability & Statistics, Logarithm, Integral Calculus, Optimization Problems, Difference and Slope, and Square. It is constructed using GeoGebra commands like Vector, Point, Circle, OrthogonalLine, Intersect, Segment, Polygon, and Line. Suitable for high school and above students learning calculus, probability statistics, and geometric construction.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Vector(B, C) → v
  3. Vector(C, D) → w
  4. Vector(D, E) → a
  5. Vector(E, F) → b
  6. Vector(F, G) → c
  7. Point(xAxis) → H
  8. Point(xAxis) → I
  9. Point(xAxis) → J
  10. Point(xAxis) → K
  11. Point(xAxis) → L
  12. Point(xAxis) → M
  13. Point(xAxis) → N
  14. Circle(H, r1) → d
  15. Circle(I, r2) → e
  16. Circle(J, r3) → g
  17. Circle(K, r4) → h
  18. Circle(L, r5) → k
  19. Circle(M, r6) → p
  20. Circle(N, r7) → q
  21. OrthogonalLine(H, xAxis) → i
  22. OrthogonalLine(I, xAxis) → j
  23. OrthogonalLine(J, xAxis) → l
  24. OrthogonalLine(K, xAxis) → m
  25. OrthogonalLine(L, xAxis) → n_1
  26. OrthogonalLine(M, xAxis) → r
  27. OrthogonalLine(N, xAxis) → s
  28. Intersect(d, i, 1) → O
  29. Segment(H, O) → t
  30. Intersect(e, j, 1) → P
  31. Segment(I, P) → f_1
  32. Intersect(g, l, 1) → Q
  33. Segment(J, Q) → g_1
  34. Intersect(h, m, 1) → R
  35. Segment(K, R) → h_1
  36. Intersect(k, n_1, 1) → S
  37. Segment(L, S) → i_1
  38. Intersect(p, r, 1) → T
  39. Segment(M, T) → j_1
  40. Intersect(q, s, 1) → U
  41. Segment(N, U) → k_1
  42. Polygon(V, W, Z) → t8, z_1, v_1, w_1
  43. Point(v_1) → A_1
  44. Point(v_1) → B_1
  45. Point(v_1) → C_1
  46. Point(v_1) → D_1
  47. Point(v_1) → E_1
  48. Line(A_1, z_1) → m_1
  49. Line(B_1, z_1) → p_1
  50. Line(C_1, z_1) → q_1
  51. Line(D_1, z_1) → r_1
  52. Line(E_1, z_1) → s_1
  53. Line(E_1, w_1) → t_1
  54. Line(D_1, w_1) → a_1
  55. Line(C_1, w_1) → b_1
  56. Line(B_1, w_1) → c_1
  57. Line(A_1, w_1) → d_1
  58. Intersect(m_1, w_1) → F_1
  59. Segment(A_1, F_1) → e_1
  60. Intersect(p_1, w_1) → G_1
  61. Segment(B_1, G_1) → f_2
  62. Intersect(q_1, w_1) → H_1
  63. Segment(C_1, H_1) → g_2
  64. Intersect(r_1, w_1) → I_1
  65. Segment(D_1, I_1) → h_2
  66. Intersect(s_1, w_1) → J_1
  67. Segment(E_1, J_1) → i_2
  68. Intersect(t_1, z_1) → K_1
  69. Segment(E_1, K_1) → j_2
  70. Intersect(a_1, z_1) → L_1
  71. Segment(D_1, L_1) → k_2
  72. Intersect(b_1, z_1) → M_1
  73. Segment(C_1, M_1) → l_2
  74. Intersect(c_1, z_1) → N_1
  75. Segment(B_1, N_1) → m_2
  76. Intersect(d_1, z_1) → O_1
  77. Segment(A_1, O_1) → n_2
  78. Line(P_1, Q_1) → p_2
  79. Line(R, p_2) → q_2
  80. Point(p_2) → R_1
  81. Point(p_2) → S_1
  82. Point(p_2) → T_1
  83. Point(p_2) → U_1
  84. Point(p_2) → V_1
  85. Point(p_2) → W_1
  86. OrthogonalLine(T_1, p_2) → r_2
  87. OrthogonalLine(U_1, p_2) → s_2
  88. OrthogonalLine(R_1, p_2) → t_2
  89. OrthogonalLine(S_1, p_2) → a_2
  90. OrthogonalLine(V_1, p_2) → b_2
  91. OrthogonalLine(W_1, p_2) → c_2
  92. Point(p_2) → Z_1
  93. Point(p_2) → A_2
  94. OrthogonalLine(Z_1, p_2) → d_2
  95. OrthogonalLine(A_2, p_2) → e_2
  96. Intersect(q_2, t_2) → B_2
  97. Intersect(q_2, a_2) → C_2
  98. Polygon(B_2, C_2, S_1, R_1) → q1, f_3, g_3, h_3, i_3
  99. Line(Q, p_2) → j_3
  100. Intersect(s_2, j_3) → D_2
  101. Intersect(t_2, j_3) → E_2
  102. Polygon(D_2, E_2, R_1, U_1) → q2, k_3, l_3, m_3, u_1
  103. Line(S, p_2) → n_3
  104. Intersect(a_2, n_3) → F_2
  105. Intersect(b_2, n_3) → G_2
  106. Polygon(F_2, G_2, V_1, S_1) → q3, p_3, q_3, r_3, s_3
  107. Line(P, p_2) → t_3
  108. Intersect(t_3, r_2) → H_2
  109. Intersect(t_3, s_2) → I_2
  110. Polygon(H_2, I_2, U_1, T_1) → q4, a_3, b_3, c_3, d_3
  111. Line(T, p_2) → e_3
  112. Intersect(b_2, e_3) → J_2
  113. Intersect(c_2, e_3) → K_2
  114. Polygon(J_2, K_2, W_1, V_1) → q5, f_4, g_4, h_4, i_4
  115. Line(O, p_2) → j_4
  116. Intersect(d_2, j_4) → L_2
  117. Intersect(r_2, j_4) → M_2
  118. Polygon(L_2, M_2, T_1, Z_1) → q6, k_4, l_4, m_4, n_4
  119. Line(U, p_2) → p_4
  120. Intersect(c_2, p_4) → N_2
  121. Intersect(e_2, p_4) → O_2
  122. Polygon(N_2, O_2, A_2, W_1) → q7, q_4, o_2, r_4, s_4