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Quarter Circle Surprise

Quarter Circle Surprise
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2017-06-01 22:07
CircleAngleProbability & StatisticsGeometric ConstructionSlider Animation
This dynamic illustration demonstrates the relationship between angles, lines, and probability & statistics problems within a quarter circle. It involves core math concepts like angles, circles, hypothesis testing, conditional probability, median value, geometric transformations, and indefinite integrals. The construction leverages GeoGebra commands such as Circle, Rotate, Angle, Line, Intersect, Ray, Segment, CircleSector, CircleArc, Dilate, Point, and Polygon. It is suitable for high school and above students to explore the integration of geometric transformations and probability statistics.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Rotate(B, 45°, A) → B'
  3. Angle(B, A, B') → α
  4. Line(B', f) → h
  5. Line(B', g) → i
  6. Intersect(f, xAxis) → D
  7. Intersect(i, f) → E
  8. Intersect(g, h) → F
  9. Line(F, B) → j
  10. Angle(B, F, B') → β
  11. Rotate(B', β, F) → B''
  12. Angle(B', F, B'') → γ
  13. Ray(F, B'') → k
  14. Intersect(c, k, 1) → G
  15. Segment(F, G) → l
  16. Circle(F, (0.5 * l)) → d
  17. Intersect(d, l, 1) → H
  18. Intersect(d, h, 2) → I
  19. CircleSector(F, I, H) → e
  20. Circle(A, (0.2 * a)) → p
  21. Ray(A, B') → m
  22. Intersect(p, g) → J, K
  23. Intersect(p, f) → L, M
  24. Intersect(p, m, 1) → N
  25. CircleSector(A, N, J) → q
  26. Angle(C, A, O) → δ
  27. Angle(B', E, O) → ε
  28. Ray(A, O) → n
  29. CircleArc(A, B, C) → r
  30. Ray(A, B) → s
  31. Ray(A, C) → t
  32. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B') → A'
  33. Segment(A, A') → f_1
  34. Point(h) → P
  35. Angle(P, F, A) → η
  36. Ray(F, P) → g_1
  37. Dilate(B', If[1 < o ≤ 2, 2 - o, 0], E) → B''_1
  38. Dilate(B', If[1 < o ≤ 2, 2 - o, 0], F) → B''_2
  39. Segment(B', B''_1) → h_1
  40. Segment(B', B''_2) → i_1
  41. Rotate(q, If[2 < o ≤ 3, ((-(o - 2)) * π / 4), ((-π)) / 4], A) → q'
  42. Circle(F, l) → c_1
  43. Intersect(c_1, j, 2) → Q
  44. Circle(Q, l) → d_1
  45. Intersect(d_1, j, 2) → R
  46. Dilate(F, If[3 < o ≤ 4, 4 - o, 0], G) → F'
  47. Segment(F, F') → j_1
  48. Dilate(F, If[4 < o < 5, 5 - o, 0], Q) → F'_1
  49. Segment(F, F'_1) → k_1
  50. Dilate(Q, If[5 < o < 6, 6 - o, 0], R) → Q'
  51. Segment(Q, Q') → l_1
  52. Dilate(R, If[6 < o < 7, 7 - o, 0], B) → R'
  53. Segment(R, R') → m_1
  54. Rotate(e, If[7 < o ≤ 8, ((-(o - 7)) * β), (-β)], F) → e'
  55. Rotate(F, If[8 < o ≤ 9, (((-(o - 8)) * 2) * β), (-2 * β)], F) → F'_2
  56. Rotate(F', If[8 < o ≤ 9, (((-(o - 8)) * 2) * β), (-2 * β)], F) → F''
  57. Segment(F'_2, F'') → n_1
  58. Rotate(F'_2, If[9 < o ≤ 10, ((-(o - 9)) * π), (-π)], F'') → S
  59. Rotate(F'', If[9 < o ≤ 10, ((-(o - 9)) * π), (-π)], F'') → F'''
  60. Segment(S, F''') → p_1
  61. Rotate(S, If[10 < o ≤ 11, ((-(o - 10)) * π), (-π)], S) → S'
  62. Rotate(F''', If[10 < o ≤ 11, ((-(o - 10)) * π), (-π)], S) → T
  63. Segment(S', T) → q_1
  64. Ray(A, U) → r_1
  65. CircleSector(A, B, C) → e_1
  66. Polygon(A, E, B', F) → poly1, a_1, e_2, b', f_2