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Derivation of Trapezium Area

Derivation of Trapezium Area
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2017-03-24 18:41
PolygonAngleTransformation
This example demonstrates the derivation of the trapezium area. Key mathematical concepts include similar triangles and isosceles triangles. Using GeoGebra commands such as Segment, Line, Point, Polygon, etc., the construction of the trapezium and related triangles is shown, illustrating the derivation of the area formula. Suitable for middle school and above students learning geometry and area calculation.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Line(C, f) → g
  3. Segment(A, C) → h_1
  4. Point(g) → D
  5. Segment(B, D) → i
  6. Polygon(A, C, D, B) → poly1, c, a, d, b
  7. Point(f) → E
  8. OrthogonalLine(E, a) → h_2
  9. Intersect(h_2, a) → F
  10. Segment(E, F) → h
  11. Vector(I, J) → v
  12. Vector(G, H) → u
  13. Translate(A, u) → A'
  14. Translate(C, u) → C'
  15. Translate(D, u) → D'
  16. Translate(B, u) → B'
  17. Polygon(A', C', D', B') → poly1', a', c', d', b'
  18. Translate(A', v) → A''
  19. Translate(C', v) → C''
  20. Translate(D', v) → D''
  21. Translate(B', v) → B''
  22. Polygon(A'', C'', D'', B'') → poly1'', a'', c'', d'', b''
  23. Translate(A, v) → A'_1
  24. Translate(C, v) → C'_1
  25. Translate(D, v) → D'_1
  26. Translate(B, v) → B'_1
  27. Polygon(A', C', B') → poly2, b'_2, a'_2, c'_2
  28. Polygon(C', D', B') → poly3, b'_3, c'_3, d'_2
  29. Rotate(B', (-Rotate), B') → B''_1
  30. Midpoint(D', B') → K
  31. Rotate(C', (-Rotate), K) → C''_1
  32. Rotate(D', (-Rotate), K) → D''_1
  33. Rotate(B', (-Rotate), K) → B''_2
  34. Polygon(C''_1, D''_1, B''_2) → poly3', b''_2, c''_1, d''_1
  35. Midpoint(A'', C'') → L
  36. Midpoint(B'', D'') → M
  37. Polygon(A'', L, M, B'') → poly4, a''_1, l, m, b''_1
  38. Polygon(L, C'', D'', M) → poly5, l_1, c''_2, d''_2, m_1
  39. Rotate(L, (-Rotate), M) → L'
  40. Rotate(C'', (-Rotate), M) → C'''
  41. Rotate(D'', (-Rotate), M) → D'''
  42. Rotate(M, (-Rotate), M) → M'
  43. Polygon(L', C''', D''', M') → poly5', l', c''', d''', m'
  44. Midpoint(A'_1, C'_1) → N
  45. Midpoint(D'_1, B'_1) → O
  46. Rotate(N, (-Rotate), N) → N'
  47. Rotate(A'_1, (-Rotate), N) → R
  48. Rotate(O, Rotate, O) → O'
  49. Rotate(B'_1, Rotate, O) → S
  50. Angle(F, E, A) → α
  51. Translate(A'', v) → A'''
  52. Translate(C'', v) → C'''_1
  53. Translate(D'', v) → D'''_1
  54. Translate(B'', v) → B'''
  55. Polygon(A''', C'''_1, D'''_1, B''') → poly1''', a''', c'''_1, d'''_1, b'''
  56. Midpoint(a''') → T
  57. Midpoint(d'''_1) → U
  58. OrthogonalLine(C'''_1, b''') → t
  59. OrthogonalLine(D'''_1, b''') → e
  60. OrthogonalLine(A''', c'''_1) → f_1
  61. OrthogonalLine(B''', c'''_1) → g_1
  62. OrthogonalLine(T, b''') → i_1
  63. OrthogonalLine(U, b''') → j_1
  64. Intersect(i_1, b''') → V
  65. Intersect(j_1, b''') → W
  66. Polygon(V, T, C'''_1, D'''_1, U, W) → poly9, v_1, t_1, c'''_2, d'''_2, u_1, w
  67. Polygon(A''', T, V) → poly10, v_2, a'''_1, t_2
  68. Polygon(W, U, B''') → poly11, b'''_1, w_1, u_2
  69. Rotate(A''', (-Rotate), T) → Z
  70. Rotate(T, (-Rotate), T) → T'
  71. Rotate(V, (-Rotate), T) → V'
  72. Polygon(Z, T', V') → poly10', v', z_1, t'
  73. Rotate(W, Rotate, U) → W'
  74. Rotate(U, Rotate, U) → U'
  75. Rotate(B''', Rotate, U) → A_1
  76. Polygon(W', U', A_1) → poly11', a_1, w', u'
  77. Intersect(i_1, c'''_1) → B_1
  78. Polygon(B_1, D'''_1, U, W, A''', T) → poly12, b_1, d'''_3, u_3, w_2, a'''_2, t_3
  79. Polygon(C'''_1, B_1, T) → poly13, t_4, c'''_3, b_2
  80. Rotate(C'''_1, Rotate, T) → C_1
  81. Rotate(B_1, Rotate, T) → D_1
  82. Rotate(T, Rotate, T) → T'_1
  83. Polygon(C_1, D_1, T'_1) → poly13', t'_1, c_1, d_1
  84. Intersect(j_1, c'''_1) → E_1
  85. Polygon(B_1, T, A''', B''', U, E_1) → poly14, b_3, t_5, a'''_3, b'''_2, u_4, e_1
  86. Polygon(U, E_1, D'''_1) → poly15, d'''_4, u_5, e_2
  87. Rotate(U, (-Rotate), U) → U'_1
  88. Rotate(E_1, (-Rotate), U) → F_1
  89. Rotate(D'''_1, (-Rotate), U) → G_1
  90. Polygon(U'_1, F_1, G_1) → poly15', g_2, u'_1, f_2
  91. Polygon(V, T, C'''_1, E_1, U, B''') → poly16, v_3, t_6, c'''_4, e_3, u_6, b'''_3
  92. PointIn(poly1) → Pointpoly1

AI Drawing Prompt Reference

Draw a trapezoid ABCD with points A, B, C, and D as its vertices. First, draw segment AB and AC, then draw a line through point C parallel to AB, intersecting at point D. Next, draw the trapezoid ABCD and its internal triangles. Then, draw a parallelogram through points A, C, and B, another through points B, D, and C, and a third through points D, B, and C. Finally, draw a fourth parallelogram through points A, C, and B, and a fifth through points A, C, D, and B, completing the construction of the trapezoid.
The prompt above was compiled from the original drawing commands and is for reference only.