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角的正交构造与识别

角的正交构造与识别
几何画板及绘图内容加载中，请稍等……
2024-04-04 21:03
角三角函数几何作图
所属专题 Besondere Winkelpaare
①本范例演示了如何构造角的正交（直角）和补角；②涉及核心数学概念：角的分类（锐角、直角、钝角）、角的度量、正交与补角关系；③作图要点：使用 GeoGebra 命令如 Polygon, PointIn, Segment, OrthogonalLine, Rotate 等绘制和变换角；④适合小学高年级学生，用于几何作图练习和角的理解。

GeoGebra 命令及参数

  1. SpeechBubbleLeft(PBubbleAlex, 10.5, 1.3) → bubble1
  2. Polygon(Q1, Q2, Q3, Q4) → q1, i_1, j_1, k_1, l
  3. PointIn(q1) → arrowCirclePoint_1
  4. Segment(arrowCircleMidpoint, arrowCirclePoint_1) → f
  5. OrthogonalLine(C, f) → g
  6. Point(g) → D
  7. Segment(D, C) → h
  8. PointIn(q1) → arrowCirclePoint
  9. Segment(arrowCircleMidpoint, arrowCirclePoint) → i
  10. OrthogonalLine(C, i) → j
  11. Point(j) → F
  12. Segment(F, C) → k
  13. Intersect(f, h) → Schnittpunkt1
  14. Intersect(i, k) → Schnittpunkt2
  15. Angle(arrowCirclePoint_1, Schnittpunkt1, D) → γ
  16. Angle(arrowCirclePoint, Schnittpunkt2, F) → δ
  17. Angle(arrowCirclePoint, arrowCircleMidpoint, arrowCirclePoint_1) → α
  18. Angle(F, C, D) → α_1
  19. Textfield(input1) → InputBox1
  20. SelectedIndex(dropdown) → index
  21. SpeechBubbleLeft(PBubbleAlex - (0, 0.1), 8.3, 1.8) → bubble2
  22. Line(arrowCircleMidpoint, arrowCirclePoint) → m
  23. Line(arrowCircleMidpoint, arrowCirclePoint_1) → n
  24. Point(n) → G
  25. Segment(arrowCirclePoint_1, G) → p
  26. Point(m) → H
  27. Segment(arrowCirclePoint, H) → q
  28. SpeechBubbleLeft(PBubbleElia, 7, 1.2) → bubble3
  29. Polygon(P1, P2, P3, P4) → p2, t_1, u, v, w
  30. PointIn(p2) → A
  31. Segment(I, A) → r
  32. Polygon(P1_{1}, P2_{1}, P3_{1}, P4_{1}) → p2_{1}, t_{2}, u_{1}, v_{1}, w_{1}
  33. PointIn(p2_{1}) → K
  34. Segment(I, K) → s
  35. Angle(K, I, A) → β
  36. Point(r) → L
  37. OrthogonalLine(L, s) → t
  38. Point(t) → M
  39. Point(t) → N
  40. Segment(M, N) → a
  41. Intersect(s, a) → O
  42. Angle(L, O, I) → ε
  43. OrthogonalLine(P, r) → b
  44. Point(b) → Q
  45. Segment(Q, P) → c
  46. Intersect(r, c) → R
  47. Angle(I, R, P) → ζ
  48. Intersect(a, c) → S
  49. Segment(M, S) → d
  50. Segment(Q, S) → e
  51. Angle(R, S, O) → η
  52. SelectedIndex(dropdown1) → index2
  53. SpeechBubbleLeft(PBubbleElia + (0.2, -0.5), 7.8, 1.3) → bubble5
  54. Textfield(input2) → InputBox2
  55. Textfield(input3) → InputBox3
  56. Ray(I, A) → f_1
  57. Point(f_1) → T
  58. Segment(A, T) → g_1
  59. Segment(arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle))), arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle))) + (arrowCircleLength * UnitOrthogonalVector[Vector[arrowCircleMidpoint, arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] + arrowCircleDistance + arrowCircleAngle)))]])) → arrowCircleArrow01_1
  60. Segment(arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle))), arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle))) - (arrowCircleLength * UnitOrthogonalVector[Vector[arrowCircleMidpoint, arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint_1] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint_1] - arrowCircleDistance - arrowCircleAngle)))]])) → arrowCircleArrow02_1
  61. Segment(arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle))), arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle))) + (arrowCircleLength * UnitOrthogonalVector[Vector[arrowCircleMidpoint, arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] + arrowCircleDistance + arrowCircleAngle)))]])) → arrowCircleArrow02
  62. Segment(arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle))), arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle))) - (arrowCircleLength * UnitOrthogonalVector[Vector[arrowCircleMidpoint, arrowCircleMidpoint + (Distance[arrowCircleMidpoint, arrowCirclePoint] * (cos(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle), sin(Angle[arrowCircleMidpoint + (1, 0), arrowCircleMidpoint, arrowCirclePoint] - arrowCircleDistance - arrowCircleAngle)))]])) → arrowCircleArrow01
  63. Ray(I, K) → h_1
  64. Point(h_1) → A_1
  65. Segment(K, A_1) → l_1
  66. PointIn(p2_{1}) → J_{1}
  67. Rotate(Segment[ArrowCenter + (ArrowDistance, 0), ArrowCenter + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((1 - 1) * (360°) / ArrowNumber), ArrowCenter) → arrow1
  68. Rotate(Segment[ArrowCenter + (ArrowDistance, 0), ArrowCenter + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((2 - 1) * (360°) / ArrowNumber), ArrowCenter) → arrow2
  69. Rotate(Segment[ArrowCenter + (ArrowDistance, 0), ArrowCenter + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((3 - 1) * (360°) / ArrowNumber), ArrowCenter) → arrow3
  70. Rotate(Segment[ArrowCenter + (ArrowDistance, 0), ArrowCenter + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((4 - 1) * (360°) / ArrowNumber), ArrowCenter) → arrow4
  71. SpeechBubbleLeft(PBubbleElia + (0.2, 0), 7.8, 1) → bubble4

AI 绘图提示语参考

绘制两个四边形用于角度演示。在第一个四边形中，从内部某点向四边形某边作两条线段，以该边上的固定点为垂足分别作这两条线段的垂线，形成四个交点，测量这些交点形成的四个角度（包括锐角、直角和钝角），同时从四边形内两个点分别向某固定点连线。在第二个四边形中，从固定点I向四边形内引两条线段，在其中一条线段上取点作垂线与另一条相交形成交点，测量由此形成的角度；在第一条线段上取点作垂线与另一条线段相交，再测量该角度。最后在四边形外部用对话气泡标注说明。
以上提示语为根据原始绘图命令整理而来，仅供参考。