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Relationship of Parallel Segment in Any Trapezoid

Relationship of Parallel Segment in Any Trapezoid
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2017-09-08 09:50
PolygonVectorCoordinate GeometryTransformationGeometric Construction
This example illustrates the relationship between a segment parallel to the bases of any trapezoid that is drawn through the intersection point. Key mathematical concepts include geometric construction and transformation. It uses commands such as Ray, Point, Segment, Dilate, Midpoint, Line, OrthogonalLine, Angle, Vector, Translate, Circle, Intersect, Rotate, Slope, Polygon. Suitable for middle school and above students to learn trapezoid properties and geometric transformations.

GeoGebra Commands & Parameters

  1. Ray(A, B) → b
  2. Point(b) → C
  3. Segment(A, C) → r
  4. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'
  5. Segment(A, A') → p
  6. Midpoint(A, C) → D
  7. Line(A, C) → c
  8. OrthogonalLine(D, c) → g
  9. Point(g) → E
  10. Ray(D, E) → h
  11. Point(h) → F
  12. Line(F, c) → l
  13. Point(l) → G
  14. Ray(F, G) → m
  15. Point(m) → I
  16. Angle(C, A, I) → δ
  17. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], I) → A'_6
  18. Vector(A, A'_6) → u_1
  19. Translate(A, u_1) → A'_7
  20. Segment(I, A) → a_1
  21. Circle(A, (0.2 * Min[a_1, r])) → c_1
  22. Intersect(c_1, r, 1) → L
  23. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], I) → A'_1
  24. Segment(A, A'_1) → q
  25. Ray(I, F) → f_2
  26. Point(f_2) → I'
  27. Angle(L, A, I') → γ
  28. Rotate(A'_7, If[11 < o ≤ 12, ((-(o - 11)) * (2 * γ)), (-2 * γ)], I) → R
  29. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], I') → A'_5
  30. Translate(A'_5, u_1) → Q
  31. Rotate(Q, If[7 < o ≤ 8, (((-(o - 7)) * 2) * γ), (-2 * γ)], I) → Q'
  32. Line(A, B) → a
  33. Slope(c) → d
  34. Slope(h) → i
  35. Point(l) → H
  36. Ray(F, H) → n
  37. Segment(C, I') → s
  38. Segment(I', I) → t
  39. Intersect(c_1, a_1, 1) → M
  40. Dilate(I, If[2 < o ≤ 3, 3 - o, 0], I') → I'_1
  41. Segment(I, I'_1) → b_1
  42. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], C) → A'_2
  43. Vector(A, A'_2) → u
  44. Translate(A, u) → A'_3
  45. Translate(A'_1, u) → N
  46. OrthogonalLine(A, r) → g_1
  47. Point(g_1) → O
  48. Angle(I, A, O) → α
  49. Angle(I', C, A) → β
  50. Rotate(A'_3, If[4 < o ≤ 5, (((o - 4) * 2) * α), (2 * α)], C) → P
  51. Rotate(N, If[4 < o ≤ 5, (((-(o - 4)) * 2) * α), (-2 * α)], C) → N'
  52. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], C) → A'_4
  53. Vector(A, A'_4) → v
  54. Dilate(I, If[7 < o ≤ 8, 8 - o, 0], I') → I'_2
  55. Vector(I, I'_2) → w
  56. Segment(I, C) → l_1
  57. Polygon(A, C, I', I) → poly1, a_2, c_2, i', i_2
  58. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], I') → C'
  59. Segment(C, C') → f_1
  60. Segment(A, I) → h_1
  61. Segment(A, C) → i_1
  62. Segment(C, I') → j_1
  63. Segment(I', I) → k_1
  64. Min({h_1, i_1, j_1, k_1}) → e_1
  65. Circle(A, (0.2 * e_1)) → p_1
  66. Intersect(p_1, i_1) → J,
  67. Rotate(J, If[4 < o ≤ 5, ((o - 4) * δ), δ], A) → J'
  68. CircleSector(A, J, J') → d_1
  69. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], I) → A'_8
  70. Vector(A, A'_8) → v_1
  71. Translate(d_1, v_1) → q_1
  72. Line(A, I) → m_1
  73. Circle(I, 2) → r_1
  74. Intersect(r_1, m_1, 2) → K
  75. Ray(I, K) → n_1
  76. Segment(A, I') → s_1
  77. Segment(I, C) → t_1
  78. Intersect(s_1, t_1) → S
  79. Line(S, p) → g_2
  80. Intersect(m_1, g_2) → T
  81. Line(I', C) → h_2
  82. Intersect(h_2, g_2) → U
  83. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], I') → A'_9
  84. Segment(A, A'_9) → j_2
  85. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], S) → C'_1
  86. Segment(C, C'_1) → k_2
  87. Dilate(S, If[8 < o ≤ 9, 9 - o, 0], I) → S'
  88. Segment(S, S') → l_2
  89. Dilate(S, If[9 < o ≤ 10, 10 - o, 0], T) → S'_1
  90. Segment(S, S'_1) → m_2
  91. Dilate(I, If[10 < o ≤ 11, 11 - o, 0], T) → I'_3
  92. Vector(I, I'_3) → w_1
  93. Translate(q_1, w_1) → d_2
  94. Dilate(S, If[9 < o ≤ 10, 10 - o, 0], U) → S'_2
  95. Segment(S, S'_2) → n_2
  96. Rotate(S, If[11 < o ≤ 12, ((o - 11) * π), π], S) → S'_3
  97. Rotate(S'_1, If[11 < o ≤ 12, ((o - 11) * π), π], S) → V
  98. Segment(S'_3, V) → p_2