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Angle Formed by 2 Chords (Dynamic)

Angle Formed by 2 Chords (Dynamic)
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2016-04-26 02:59
CircleAngleTransformationGeometric ConstructionSlider Animation
This GeoGebra example dynamically illustrates the geometric property of the angle formed by two intersecting chords inside a circle. The core mathematical concept is the corollary of the inscribed angle theorem: the measure of the angle formed by two chords intersecting inside a circle equals half the sum of the measures of the intercepted arcs. The construction uses commands such as Circle, Segment, Point, Intersect, Angle, Rotate, Dilate, Translate, and CircleSector to build the circle, chords, and their intersection, with sliders enabling interactive manipulation. This resource is ideal for middle and high school geometry classes, helping students visualize and explore the relationship between angles and arcs in a circle through dynamic dragging and observation.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Segment(A, B) → a
  3. Point(c) → C
  4. Point(c) → D
  5. Segment(C, D) → d
  6. Point(d) → E
  7. CircleArc(A, C, D) → e
  8. Point(e) → F
  9. CircleArc(A, D, C) → f
  10. Ray(F, E) → b
  11. Intersect(f, b, 1) → G
  12. Segment(F, E) → g
  13. Segment(E, G) → h
  14. Segment(C, E) → i
  15. Segment(E, D) → j
  16. Min(list1) → k
  17. Circle(E, (0.4 * k)) → p
  18. Segment(A, G) → l
  19. Segment(A, C) → m
  20. Segment(A, F) → n
  21. Segment(A, D) → q
  22. Circle(A, (0.4 * k)) → r
  23. Intersect(r, l, 1) → H
  24. Intersect(r, m, 1) → I
  25. Intersect(p, h, 1) → J
  26. Intersect(p, i, 1) → K
  27. Intersect(p, g, 1) → L
  28. Intersect(p, j, 1) → M
  29. Intersect(r, n, 1) → N
  30. Intersect(r, q, 1) → O
  31. Angle(G, A, C) → α
  32. Angle(F, A, D) → β
  33. Rotate(G, (-(α / 2)), A) → G'
  34. Rotate(G, α / 2, A) → G'_1
  35. Angle(G, A, G'_1) → γ
  36. Rotate(F, β / 2, A) → F'
  37. Angle(F, A, F') → δ
  38. Segment(A, F') → s
  39. Segment(A, G'_1) → t
  40. Intersect(r, t, 1) → P
  41. Intersect(r, s, 1) → Q
  42. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], G) → A'
  43. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'_1
  44. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], F) → A'_2
  45. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'_3
  46. Segment(A, A') → a_1
  47. Segment(A, A'_1) → b_1
  48. Segment(A, A'_2) → c_1
  49. Segment(A, A'_3) → d_1
  50. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], G'_1) → A'_4
  51. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], F') → A'_5
  52. Segment(A, A'_4) → e_1
  53. Segment(A, A'_5) → f_1
  54. CircleSector(A, H, P) → g_1
  55. CircleSector(A, N, Q) → h_1
  56. Rotate(g_1, If[2 < o ≤ 3, ((o - 2) * γ), γ], A) → k_1
  57. Rotate(h_1, If[3 < o ≤ 4, ((o - 3) * δ), δ], A) → q_1
  58. Dilate(A, If[4 < o ≤ 5, 5 - o, 0], E) → A'_6
  59. Vector(A, A'_6) → u
  60. Translate(k_1, u) → s_1
  61. Translate(I, u) → I'
  62. Angle(I', E, K) → ε
  63. Rotate(s_1, If[5 < o ≤ 6, ((o - 5) * ε), ε], E) → t_1
  64. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], E) → A'_7
  65. Vector(A, A'_7) → v
  66. Translate(q_1, v) → c_2
  67. Angle(J, E, K) → ζ
  68. Translate(P, v) → P'
  69. Rotate(P', ε, E) → P''
  70. Translate(O, v) → O'
  71. Angle(O', E, P'') → η
  72. Rotate(c_2, If[7 < o ≤ 8, ((o - 7) * η), η], E) → d_2
  73. CircleArc(A, F, D) → e_2
  74. CircleArc(A, G, C) → f_2
  75. Intersect(p, h, 1) → R
  76. CircleSector(E, R, K) → g_2