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Diagonal Problem

Diagonal Problem
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2020-08-18 20:54
TrianglePolygonTrigonometryCalculusProbability & Statistics
This example demonstrates constructing diagonals in rectangles, squares, and triangles using dynamic geometry, exploring trigonometric relationships and the probability distribution of diagonal lengths, and evaluating the indefinite integral of the hypotenuse lengths. Core concepts include geometric construction, orthocenter, sine and cosine laws, trigonometric identities, and integration. The GeoGebra construction uses Polygon, Point, and Segment commands to create multiple polygons and points, allowing interactive dragging to observe changes in real time. It is suitable for high school or early university courses in geometry and calculus.

GeoGebra Commands & Parameters

  1. Polygon((1, 0), (2, 0), (2, 1), (1, 1)) → poly2, e, f, g, h
  2. Polygon((2, 0), (3, 0), (3, 1), (2, 1)) → poly3, i, j, k, l
  3. Polygon((3, 0), (4, 0), (4, 1), (3, 1)) → poly4, m, n, p, q
  4. Polygon((4, 0), (5, 0), (5, 1), (4, 1)) → poly5, r, s, t, a_1
  5. Polygon((5, 0), (6, 0), (6, 1), (5, 1)) → poly6, b_1, c_1, d_1, e_1
  6. Polygon((6, 0), (7, 0), (7, 1), (6, 1)) → poly7, f_1, g_1, h_1, i_1
  7. Polygon((7, 0), (8, 0), (8, 1), (7, 1)) → poly8, j_1, k_1, l_1, m_1
  8. Polygon((8, 0), (9, 0), (9, 1), (8, 1)) → poly9, n_1, p_1, q_1, r_1
  9. Polygon((9, 0), (10, 0), (10, 1), (9, 1)) → poly10, s_1, t_1, a_2, b_2
  10. Polygon((10, 0), (11, 0), (11, 1), (10, 1)) → poly11, c_2, d_2, e_2, f_2
  11. Polygon((11, 0), (12, 0), (12, 1), (11, 1)) → poly12, g_2, h_2, i_2, j_2
  12. Polygon((12, 0), (13, 0), (13, 1), (12, 1)) → poly13, k_2, l_2, m_2, n_2
  13. Polygon((13, 0), (14, 0), (14, 1), (13, 1)) → poly14, p_2, q_2, r_2, s_2
  14. Polygon((14, 0), (15, 0), (15, 1), (14, 1)) → poly15, t_2, a_3, b_3, c_3
  15. Polygon((0, 1), (1, 1), (1, 2), (0, 2)) → poly16, d_3, e_3, f_3, g_3
  16. Polygon((1, 1), (2, 1), (2, 2), (1, 2)) → poly17, h_3, i_3, j_3, k_3
  17. Polygon((2, 1), (3, 1), (3, 2), (2, 2)) → poly18, l_3, m_3, n_3, p_3
  18. Polygon((3, 1), (4, 1), (4, 2), (3, 2)) → poly19, q_3, r_3, s_3, t_3
  19. Polygon((4, 1), (5, 1), (5, 2), (4, 2)) → poly20, a_4, b_4, c_4, d_4
  20. Polygon((5, 1), (6, 1), (6, 2), (5, 2)) → poly21, e_4, f_4, g_4, h_4
  21. Polygon((6, 1), (7, 1), (7, 2), (6, 2)) → poly22, i_4, j_4, k_4, l_4
  22. Polygon((7, 1), (8, 1), (8, 2), (7, 2)) → poly23, m_4, n_4, p_4, q_4
  23. Polygon((8, 0.9999999999999996), (9, 0.9999999999999996), (9, 1.9999999999999996), (8, 1.9999999999999996)) → poly24, r_4, s_4, t_4, a_5
  24. Polygon((9, 1), (10, 1), (10, 2), (9, 2)) → poly25, b_5, c_5, d_5, e_5
  25. Polygon((10, 1), (11, 1), (11, 2), (10, 2)) → poly26, f_5, g_5, h_5, i_5
  26. Polygon((11, 1), (12, 1), (12, 2), (11, 2)) → poly27, j_5, k_5, l_5, m_5
  27. Polygon((12, 1), (13, 1), (13, 2), (12, 2)) → poly28, n_5, p_5, q_5, r_5
  28. Polygon((13, 1), (14, 1), (14, 2), (13, 2)) → poly29, s_5, t_5, a_6, b_6
  29. Polygon((14, 1), (15, 1), (15, 2), (14, 2)) → poly30, c_6, d_6, e_6, f_6
  30. Polygon((0, 2), (1, 2), (1, 3), (0, 3)) → poly31, g_6, h_6, i_6, j_6
  31. Polygon((1, 2), (2, 2), (2, 3), (1, 3)) → poly32, k_6, l_6, m_6, n_6
  32. Polygon((2, 2), (3, 2), (3, 3), (2, 3)) → poly33, p_6, q_6, r_6, s_6
  33. Polygon((3, 2), (4, 2), (4, 3), (3, 3)) → poly34, t_6, a_7, b_7, c_7
  34. Polygon((4, 2), (5, 2), (5, 3), (4, 3)) → poly35, d_7, e_7, f_7, g_7
  35. Polygon((5, 2), (6, 2), (6, 3), (5, 3)) → poly36, h_7, i_7, j_7, k_7
  36. Polygon((6, 2), (7, 2), (7, 3), (6, 3)) → poly37, l_7, m_7, n_7, p_7
  37. Polygon((7, 2), (8, 2), (8, 3), (7, 3)) → poly38, q_7, r_7, s_7, t_7
  38. Polygon((8, 2), (9, 2), (9, 3), (8, 3)) → poly39, a_8, b_8, c_8, d_8
  39. Polygon((9, 2), (10, 2), (10, 3), (9, 3)) → poly40, e_8, f_8, g_8, h_8
  40. Polygon((10, 2), (11, 2), (11, 3), (10, 3)) → poly41, i_8, j_8, k_8, l_8
  41. Polygon((11, 2), (12, 2), (12, 3), (11, 3)) → poly42, m_8, n_8, p_8, q_8
  42. Polygon((11.999999999999998, 2), (12.999999999999998, 2), (12.999999999999998, 3), (11.999999999999998, 3)) → poly43, r_8, s_8, t_8, a_9
  43. Polygon((13, 2), (14, 2), (14, 3), (13, 3)) → poly44, b_9, c_9, d_9, e_9
  44. Polygon((14, 2), (15, 2), (15, 3), (14, 3)) → poly45, f_9, g_9, h_9, i_9
  45. Polygon((0, 3), (1, 3), (1, 4), (0, 4)) → poly46, j_9, k_9, l_9, m_9
  46. Polygon((1, 3), (2, 3), (2, 4), (1, 4)) → poly47, n_9, p_9, q_9, r_9
  47. Polygon((2, 3), (3, 3), (3, 4), (2, 4)) → poly48, s_9, t_9, a_{10}, b_{10}
  48. Polygon((3, 3), (4, 3), (4, 4), (3, 4)) → poly49, c_{10}, d_{10}, e_{10}, f_{10}
  49. Polygon((4, 3), (5, 3), (5, 4), (4, 4)) → poly50, g_{10}, h_{10}, i_{10}, j_{10}
  50. Polygon((5, 3), (6, 3), (6, 4), (5, 4)) → poly51, k_{10}, l_{10}, m_{10}, n_{10}
  51. Polygon((6, 2.999999999999999), (7, 2.999999999999999), (7, 3.999999999999999), (6, 3.999999999999999)) → poly52, p_{10}, q_{10}, r_{10}, s_{10}
  52. Polygon((7, 3), (8, 3), (8, 4), (7, 4)) → poly53, t_{10}, a_{11}, b_{11}, c_{11}
  53. Polygon((8, 3), (9, 3), (9, 4), (8, 4)) → poly54, d_{11}, e_{11}, f_{11}, g_{11}
  54. Polygon((9, 3), (10, 3), (10, 4), (9, 4)) → poly55, h_{11}, i_{11}, j_{11}, k_{11}
  55. Polygon((10, 3), (11, 3), (11, 4), (10, 4)) → poly56, l_{11}, m_{11}, n_{11}, p_{11}
  56. Polygon((11, 3), (12, 3), (12, 4), (11, 4)) → poly57, q_{11}, r_{11}, s_{11}, t_{11}
  57. Polygon((12, 3), (13, 3), (13, 4), (12, 4)) → poly58, a_{12}, b_{12}, c_{12}, d_{12}
  58. Polygon((13, 3), (14, 3), (14, 4), (13, 4)) → poly59, e_{12}, f_{12}, g_{12}, h_{12}
  59. Polygon((13.999999999999996, 3), (14.999999999999996, 3), (14.999999999999996, 4), (13.999999999999996, 4)) → poly60, i_{12}, j_{12}, k_{12}, l_{12}
  60. Polygon((0, 4), (1, 4), (1, 5), (0, 5)) → poly61, m_{12}, n_{12}, p_{12}, q_{12}
  61. Polygon((1, 4), (2, 4), (2, 5), (1, 5)) → poly62, r_{12}, s_{12}, t_{12}, a_{13}
  62. Polygon((3, 4), (4, 4), (4, 5), (3, 5)) → poly64, f_{13}, g_{13}, h_{13}, i_{13}
  63. Polygon((4, 4), (5, 4), (5, 5), (4, 5)) → poly65, j_{13}, k_{13}, l_{13}, m_{13}
  64. Polygon((5, 4), (6, 4), (6, 5), (5, 5)) → poly66, n_{13}, p_{13}, q_{13}, r_{13}
  65. Polygon((6, 4), (7, 4), (7, 5), (6, 5)) → poly67, s_{13}, t_{13}, a_{14}, b_{14}
  66. Polygon((7, 4), (8, 4), (8, 5), (7, 5)) → poly68, c_{14}, d_{14}, e_{14}, f_{14}
  67. Polygon((8, 4), (9, 4), (9, 5), (8, 5)) → poly69, g_{14}, h_{14}, i_{14}, j_{14}
  68. Polygon((9, 4), (10, 4), (10, 5), (9, 5)) → poly70, k_{14}, l_{14}, m_{14}, n_{14}
  69. Polygon((9.999999999999998, 4), (10.999999999999998, 4), (10.999999999999998, 5), (9.999999999999998, 5)) → poly71, p_{14}, q_{14}, r_{14}, s_{14}
  70. Polygon((11, 4), (12, 4), (12, 5), (11, 5)) → poly72, t_{14}, a_{15}, b_{15}, c_{15}
  71. Polygon((12, 4), (13, 4), (13, 5), (12, 5)) → poly73, d_{15}, e_{15}, f_{15}, g_{15}
  72. Polygon((13, 4), (14, 4), (14, 5), (13, 5)) → poly74, h_{15}, i_{15}, j_{15}, k_{15}
  73. Polygon((14, 4), (15, 4), (15, 5), (14, 5)) → poly75, l_{15}, m_{15}, n_{15}, p_{15}
  74. Polygon((0, 5), (1, 5), (1, 6), (0, 6)) → poly76, q_{15}, r_{15}, s_{15}, t_{15}
  75. Polygon((1, 5), (2, 5), (2, 6), (1, 6)) → poly77, a_{16}, b_{16}, c_{16}, d_{16}
  76. Polygon((2, 5), (3, 5), (3, 6), (2, 6)) → poly78, e_{16}, f_{16}, g_{16}, h_{16}
  77. Polygon((3, 5), (4, 5), (4, 6), (3, 6)) → poly79, i_{16}, j_{16}, k_{16}, l_{16}
  78. Polygon((4, 5), (5, 5), (5, 6), (4, 6)) → poly80, m_{16}, n_{16}, p_{16}, q_{16}
  79. Polygon((5, 4.999999999999998), (6, 4.999999999999998), (6, 5.999999999999998), (5, 5.999999999999998)) → poly81, r_{16}, s_{16}, t_{16}, a_{17}
  80. Polygon((6, 5), (7, 5), (7, 6), (6, 6)) → poly82, b_{17}, c_{17}, d_{17}, e_{17}
  81. Polygon((7, 5), (8, 5), (8, 6), (7, 6)) → poly83, f_{17}, g_{17}, h_{17}, i_{17}
  82. Polygon((8, 5), (9, 5), (9, 6), (8, 6)) → poly84, j_{17}, k_{17}, l_{17}, m_{17}
  83. Polygon((9, 4.999999999999998), (10, 4.999999999999998), (10, 5.999999999999998), (9, 5.999999999999998)) → poly85, n_{17}, p_{17}, q_{17}, r_{17}
  84. Polygon((10, 5), (11, 5), (11, 6), (10, 6)) → poly86, s_{17}, t_{17}, a_{18}, b_{18}
  85. Polygon((11, 5), (12, 5), (12, 6), (11, 6)) → poly87, c_{18}, d_{18}, e_{18}, f_{18}
  86. Polygon((12, 5), (13, 5), (13, 6), (12, 6)) → poly88, g_{18}, h_{18}, i_{18}, j_{18}
  87. Polygon((13, 5), (14, 5), (14, 6), (13, 6)) → poly89, k_{18}, l_{18}, m_{18}, n_{18}
  88. Polygon((14, 5), (15, 5), (15, 6), (14, 6)) → poly90, p_{18}, q_{18}, r_{18}, s_{18}
  89. Polygon((0, 6), (1, 6), (1, 7), (0, 7)) → poly91, t_{18}, a_{19}, b_{19}, c_{19}
  90. Polygon((1, 6), (2, 6), (2, 7), (1, 7)) → poly92, d_{19}, e_{19}, f_{19}, g_{19}
  91. Polygon((2, 6), (3, 6), (3, 7), (2, 7)) → poly93, h_{19}, i_{19}, j_{19}, k_{19}
  92. Polygon((3, 6), (4, 6), (4, 7), (3, 7)) → poly94, l_{19}, m_{19}, n_{19}, p_{19}
  93. Polygon((4, 6), (5, 6), (5, 7), (4, 7)) → poly95, q_{19}, r_{19}, s_{19}, t_{19}
  94. Polygon((5, 6), (6, 6), (6, 7), (5, 7)) → poly96, a_{20}, b_{20}, c_{20}, d_{20}
  95. Polygon((6, 5.999999999999998), (7, 5.999999999999998), (7, 6.999999999999998), (6, 6.999999999999998)) → poly97, e_{20}, f_{20}, g_{20}, h_{20}
  96. Polygon((7, 6), (8, 6), (8, 7), (7, 7)) → poly98, i_{20}, j_{20}, k_{20}, l_{20}
  97. Polygon((8, 5.999999999999998), (9, 5.999999999999998), (9, 6.999999999999998), (8, 6.999999999999998)) → poly99, m_{20}, n_{20}, p_{20}, q_{20}
  98. Polygon((9, 6), (10, 6), (10, 7), (9, 7)) → poly100, r_{20}, s_{20}, t_{20}, a_{21}
  99. Polygon((10, 6), (11, 6), (11, 7), (10, 7)) → poly101, b_{21}, c_{21}, d_{21}, e_{21}
  100. Polygon((11, 6), (12, 6), (12, 7), (11, 7)) → poly102, f_{21}, g_{21}, h_{21}, i_{21}
  101. Polygon((12, 6), (13, 6), (13, 7), (12, 7)) → poly103, j_{21}, k_{21}, l_{21}, m_{21}
  102. Polygon((13, 6), (14, 6), (14, 7), (13, 7)) → poly104, n_{21}, p_{21}, q_{21}, r_{21}
  103. Polygon((14, 6), (15, 6), (15, 7), (14, 7)) → poly105, s_{21}, t_{21}, a_{22}, b_{22}
  104. Polygon((0, 7), (1, 7), (1, 8), (0, 8)) → poly106, c_{22}, d_{22}, e_{22}, f_{22}
  105. Polygon((1, 7), (2, 7), (2, 8), (1, 8)) → poly107, g_{22}, h_{22}, i_{22}, j_{22}
  106. Polygon((2, 7), (3, 7), (3, 8), (2, 8)) → poly108, k_{22}, l_{22}, m_{22}, n_{22}
  107. Polygon((3, 7), (4, 7), (4, 8), (3, 8)) → poly109, p_{22}, q_{22}, r_{22}, s_{22}
  108. Polygon((4, 7), (5, 7), (5, 8), (4, 8)) → poly110, t_{22}, a_{23}, b_{23}, c_{23}
  109. Polygon((5, 7), (6, 7), (6, 8), (5, 8)) → poly111, d_{23}, e_{23}, f_{23}, g_{23}
  110. Polygon((6, 7), (7, 7), (7, 8), (6, 8)) → poly112, h_{23}, i_{23}, j_{23}, k_{23}
  111. Polygon((7, 7), (8, 7), (8, 8), (7, 8)) → poly113, l_{23}, m_{23}, n_{23}, p_{23}
  112. Polygon((8, 7), (9, 7), (9, 8), (8, 8)) → poly114, q_{23}, r_{23}, s_{23}, t_{23}
  113. Polygon((9, 7), (10, 7), (10, 8), (9, 8)) → poly115, a_{24}, b_{24}, c_{24}, d_{24}
  114. Polygon((9.999999999999998, 7), (10.999999999999998, 7), (10.999999999999998, 8), (9.999999999999998, 8)) → poly116, e_{24}, f_{24}, g_{24}, h_{24}
  115. Polygon((11, 6.999999999999998), (12, 6.999999999999998), (12, 7.999999999999998), (11, 7.999999999999998)) → poly117, i_{24}, j_{24}, k_{24}, l_{24}
  116. Polygon((12, 7), (13, 7), (13, 8), (12, 8)) → poly118, m_{24}, n_{24}, p_{24}, q_{24}
  117. Polygon((13, 7), (14, 7), (14, 8), (13, 8)) → poly119, r_{24}, s_{24}, t_{24}, a_{25}
  118. Polygon((14, 7), (15, 7), (15, 8), (14, 8)) → poly120, b_{25}, c_{25}, d_{25}, e_{25}
  119. Polygon((0, 8), (1, 8), (1, 9), (0, 9)) → poly121, f_{25}, g_{25}, h_{25}, i_{25}
  120. Polygon((1, 8), (2, 8), (2, 9), (1, 9)) → poly122, j_{25}, k_{25}, l_{25}, m_{25}
  121. Polygon((2, 8), (3, 8), (3, 9), (2, 9)) → poly123, n_{25}, p_{25}, q_{25}, r_{25}
  122. Polygon((3, 8), (4, 8), (4, 9), (3, 9)) → poly124, s_{25}, t_{25}, a_{26}, b_{26}
  123. Polygon((4, 8), (5, 8), (5, 9), (4, 9)) → poly125, c_{26}, d_{26}, e_{26}, f_{26}
  124. Polygon((5, 8), (6, 8), (6, 9), (5, 9)) → poly126, g_{26}, h_{26}, i_{26}, j_{26}
  125. Polygon((6, 8), (7, 8), (7, 9), (6, 9)) → poly127, k_{26}, l_{26}, m_{26}, n_{26}
  126. Polygon((7, 7.9999999999999964), (8, 7.9999999999999964), (8, 8.999999999999996), (7, 8.999999999999996)) → poly128, p_{26}, q_{26}, r_{26}, s_{26}
  127. Polygon((8, 8), (9, 8), (9, 9), (8, 9)) → poly129, t_{26}, a_{27}, b_{27}, c_{27}
  128. Polygon((9, 8), (10, 8), (10, 9), (9, 9)) → poly130, d_{27}, e_{27}, f_{27}, g_{27}
  129. Polygon((10, 8), (11, 8), (11, 9), (10, 9)) → poly131, h_{27}, i_{27}, j_{27}, k_{27}
  130. Polygon((11, 8), (12, 8), (12, 9), (11, 9)) → poly132, l_{27}, m_{27}, n_{27}, p_{27}
  131. Polygon((12, 8), (13, 8), (13, 9), (12, 9)) → poly133, q_{27}, r_{27}, s_{27}, t_{27}
  132. Polygon((13, 8), (14, 8), (14, 9), (13, 9)) → poly134, a_{28}, b_{28}, c_{28}, d_{28}
  133. Polygon((14, 8), (15, 8), (15, 9), (14, 9)) → poly135, e_{28}, f_{28}, g_{28}, h_{28}
  134. Polygon((0, 9), (1, 9), (1, 10), (0, 10)) → poly136, i_{28}, j_{28}, k_{28}, l_{28}
  135. Polygon((1, 9), (2, 9), (2, 10), (1, 10)) → poly137, m_{28}, n_{28}, p_{28}, q_{28}
  136. Polygon((2, 9), (3, 9), (3, 10), (2, 10)) → poly138, r_{28}, s_{28}, t_{28}, a_{29}
  137. Polygon((3, 9), (4, 9), (4, 10), (3, 10)) → poly139, b_{29}, c_{29}, d_{29}, e_{29}
  138. Polygon((4, 9), (5, 9), (5, 10), (4, 10)) → poly140, f_{29}, g_{29}, h_{29}, i_{29}
  139. Polygon((5, 9), (6, 9), (6, 10), (5, 10)) → poly141, j_{29}, k_{29}, l_{29}, m_{29}
  140. Polygon((6, 9), (7, 9), (7, 10), (6, 10)) → poly142, n_{29}, p_{29}, q_{29}, r_{29}
  141. Polygon((7, 9), (8, 9), (8, 10), (7, 10)) → poly143, s_{29}, t_{29}, a_{30}, b_{30}
  142. Polygon((8, 9), (9, 9), (9, 10), (8, 10)) → poly144, c_{30}, d_{30}, e_{30}, f_{30}
  143. Polygon((9, 9), (10, 9), (10, 10), (9, 10)) → poly145, g_{30}, h_{30}, i_{30}, j_{30}
  144. Polygon((10, 9), (11, 9), (11, 10), (10, 10)) → poly146, k_{30}, l_{30}, m_{30}, n_{30}
  145. Polygon((11, 9), (12, 9), (12, 10), (11, 10)) → poly147, p_{30}, q_{30}, r_{30}, s_{30}
  146. Polygon((12, 9), (13, 9), (13, 10), (12, 10)) → poly148, t_{30}, a_{31}, b_{31}, c_{31}
  147. Polygon((13, 9), (14, 9), (14, 10), (13, 10)) → poly149, d_{31}, e_{31}, f_{31}, g_{31}
  148. Polygon((14, 8.999999999999996), (15, 8.999999999999996), (15, 9.999999999999996), (14, 9.999999999999996)) → poly150, h_{31}, i_{31}, j_{31}, k_{31}
  149. Polygon((2, 4), (3, 4), (3, 5), (2, 5)) → poly63, b_{13}, c_{13}, d_{13}, e_{13}
  150. Polygon((0, 0), (1, 0), (1, 1), (0, 1)) → poly1, a, b, c, d
  151. Point(Sequence[Sequence[(i, j), i, 0, 15], j, 0, 10]) → P
  152. Segment((0, 0), (x(P), 0)) → l_{31}
  153. Segment((x(P), 0), P) → m_{31}
  154. Segment(P, (0, y(P))) → n_{31}
  155. Segment((0, 0), (0, y(P))) → p_{31}
  156. Segment((0, 0), P) → q_{31}

AI Drawing Prompt Reference

The figure consists of four rows of unit squares aligned to the axes: the first row contains 14 squares spanning x=1 to 15, the second and third rows each contain 15 squares spanning x=0 to 15, and the fourth row contains 4 squares spanning x=0 to 4. All 48 squares share edges to form a stepped rectangular arrangement with a missing left block in the top row and a truncated right side in the bottom row.
The prompt above was compiled from the original drawing commands and is for reference only.