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Area of Trapezium -- Duplication: Dynamic Geometric Construction

Area of Trapezium -- Duplication: Dynamic Geometric Construction
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2017-12-11 14:31
PolygonCoordinate GeometryTransformationGeometric ConstructionSlider Animation
In collections Mathematics Form 1
This GeoGebra example demonstrates the area of a trapezium through a duplication construction, visually showing how the original trapezoid can be transformed into a shape with twice its area, such as a parallelogram, to verify the area formula. The core mathematical concepts include trapezium (trapezoid), parallelogram, area conservation, rotation transformation, midpoints, and inscribed circles. The construction uses GeoGebra commands such as Line, Point, Polygon, DynamicCoordinates, SelectedIndex, Midpoint, Circle, Angle, Intersect, and rotation (Putar) to create the original trapezoid and its duplicated copies (poly1', poly1''), along with auxiliary points and lines. This resource is suitable for middle school or high school geometry classes, helping students intuitively understand the derivation of the trapezoid area formula and serving as a model for dynamic geometry lesson design.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → a
  2. Point(a) → D
  3. If(abs(x(H) - x(A)) < 0.2, (x(A), y(H)), H) → Pm
  4. DynamicCoordinates(H, x(Pm), y(Pm)) → B
  5. Line(B, a) → c
  6. Point(c) → C
  7. If(p ≟ true, C, D) → Putar
  8. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d
  9. Line(A, yAxis) → b
  10. Intersect(c, b) → E
  11. If(e ≤ 1, Rotate[B, (180° * e), Putar], Rotate[B, 180°, Putar]) → B'
  12. If(e ≤ 1, Rotate[C, (180° * e), Putar], Rotate[C, 180°, Putar]) → C'
  13. If(e ≤ 1, Rotate[D, (180° * e), Putar], Rotate[D, 180°, Putar]) → D'
  14. If(e ≤ 1, Rotate[A, (180° * e), Putar], Rotate[A, 180°, Putar]) → A'
  15. If(p ≟ true, Vector[C, D], Vector[D, C]) → u
  16. Polygon(A', B', C', D') → poly1', a', b', c', d'
  17. If(e > 1, Translate[A', Vector[((e - 1) * u)]]) → A''
  18. If(e > 1, Translate[B', Vector[((e - 1) * u)]]) → B''
  19. If(e > 1, Translate[C', Vector[((e - 1) * u)]]) → C''
  20. If(e > 1, Translate[D', Vector[((e - 1) * u)]]) → D''
  21. Polygon(A'', B'', C'', D'') → poly1'', a'', b'', c'', d''
  22. Midpoint(B, C) → F
  23. Midpoint(A, D) → G
  24. Circle(F, ja) → f
  25. Intersect(f, c) → I_2, I_1
  26. Circle(I_1, ja) → k
  27. Circle(I_2, ja) → p_2
  28. Line(I_1, b) → j
  29. Line(I_2, b) → g
  30. Intersect(p_2, g, 2) → I_3
  31. Intersect(k, j, 2) → I_4
  32. Circle(C, ja) → q
  33. Line(C, b) → i
  34. Intersect(q, i, 2) → L
  35. Circle(B, ja) → r
  36. Line(B, b) → l
  37. Intersect(r, l, 2) → M
  38. Circle(G, ja) → s
  39. Intersect(s, a) → P_4, P_3
  40. Circle(P_3, ja) → t
  41. Circle(P_4, ja) → d_1
  42. Line(P_3, b) → h
  43. Line(P_4, b) → m
  44. Intersect(d_1, m, 1) → P_1
  45. Intersect(t, h, 1) → P_2
  46. Circle(D, ja) → e_1
  47. Circle(A, ja) → f_1
  48. Intersect(f_1, b, 1) → R
  49. Line(D, b) → n
  50. Intersect(e_1, n, 1) → S
  51. Circle(E, ja) → g_1
  52. Intersect(a, i) → T
  53. Intersect(g_1, c, 1) → U
  54. Intersect(f_1, a, 1) → V
  55. Midpoint(E, A) → W
  56. Circle(W, ja) → h_1
  57. Intersect(h_1, b, 2) → Z
  58. Intersect(h_1, b, 1) → A_1
  59. Circle(Z, ja) → k_1
  60. Circle(A_1, ja) → p_1
  61. Line(Z, c) → i_1
  62. Line(A_1, c) → j_1
  63. Intersect(k_1, i_1, 1) → B_1
  64. Intersect(p_1, j_1, 1) → C_1
  65. Intersect(l, i_1) → D_1
  66. Intersect(l, j_1) → E_1
  67. Intersect(a, l) → F_1
  68. Circle(D_1, ja) → q_1
  69. Circle(E_1, ja) → r_1
  70. Midpoint(F_1, B) → G_1
  71. Intersect(r, c, 1) → H_1
  72. Intersect(q_1, i_1, 1) → I_5
  73. Intersect(r_1, j_1, 1) → J_1
  74. Circle(F_1, ja) → s_1
  75. Intersect(s_1, a, 1) → K_1
  76. Intersect(r_1, j_1, 2) → I
  77. Intersect(q_1, i_1, 2) → J
  78. Intersect(s_1, a, 2) → K
  79. Intersect(r, c, 2) → N
  80. Intersect(k, j, 1) → O
  81. Intersect(q, i, 1) → P
  82. Intersect(p_2, g, 1) → Q
  83. Intersect(r, l, 1) → L_1
  84. If(p ≟ true, Vector[I_3, M], Vector[Q, L_1]) → v
  85. If(p ≟ true, Vector[I_4, L], Vector[O, P]) → w
  86. Intersect(f_1, b, 2) → M_1
  87. Intersect(d_1, m, 2) → N_1
  88. Intersect(t, h, 2) → O_1
  89. Intersect(e_1, n, 2) → Q_1
  90. If(p ≟ true, Vector[P_1, R], Vector[N_1, M_1]) → u_1
  91. If(p ≟ true, Vector[P_2, S], Vector[O_1, Q_1]) → v_1
  92. If(p ≟ true, Vector[B_1, V], Vector[C_1, V]) → w_1
  93. If(p ≟ true, Vector[C_1, U], Vector[B_1, U]) → l_1
  94. If(p ≟ true, Vector[J, K], Vector[I, K]) → m_1
  95. If(p ≟ true, Vector[I, N], Vector[J, N]) → n_1
  96. Angle(D, A, B) → α
  97. Angle(C, B, A) → β
  98. Angle(C, D, A) → γ
  99. Angle(D, C, B) → δ
  100. SelectedIndex(Bahasa) → nn
  101. If(nn ≟ 1, "Derive the formula to calculate", "Menerbit rumus untuk menghitungkan") → Tex1
  102. If(nn ≟ 1, "the area of a trapezium", "luas bentuk trapezium") → Tex2

AI Drawing Prompt Reference

Draw a trapezoid with a horizontal base. Place two points on the base as lower vertices, then draw a parallel line above and place two points as upper vertices, connecting all four to form the trapezoid. Rotate the entire trapezoid 180 degrees around one of its vertices to create a rotated copy. Then translate this rotated copy horizontally so that it joins the original trapezoid to form a parallelogram. Finally, draw the diagonals of the trapezoid, mark their midpoints, and connect these midpoints with dashed lines to show the area relationship.
The prompt above was compiled from the original drawing commands and is for reference only.