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Interactive Trapezoid Area Discovery

Interactive Trapezoid Area Discovery
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2018-11-13 10:03
TrigonometryCoordinate GeometryGeometric Construction
In collections Perimeter & Area
This example demonstrates how to explore the area of a trapezoid through interactive manipulation. Core math concepts include the area formula for trapezoids, properties of the median line, logical reasoning, and applications of the Pythagorean theorem and trigonometry. Construction highlights involve using commands and objects like Ray, Point, Segment, Slope, PointIn, Midpoint, Line, Dilate, Rotate, Polygon, OrthogonalLine, Intersect, Min, Circle, Angle, CircleSector, Vector, Translate in GeoGebra. Suitable for middle school and above students to understand area calculation and geometric constructions.

GeoGebra Commands & Parameters

  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