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矩形构造：中间开放问题

矩形构造：中间开放问题
几何画板及绘图内容加载中，请稍等……
2021-01-08 21:35
多边形角解析几何几何作图
所属专题 Perimeter & Area Elementary Math Resources
本例演示如何利用作图约束来构造矩形（含正方形）并分析“中间条件未封闭”的开放式几何问题。核心数学概念包括：矩形与正方形的性质、面积与周长的联系、相似三角形用于建立平行与垂直关系、二维向量/坐标关系帮助定位顶点，以及毕达哥拉斯定理用于校验边长与对角线的长度关系。作图要点：从构造摘要可见，主要使用 GeoGebra 的 Polygon 与 Textfield 命令来逐步生成多种顶点集合 poly1, poly1_1, poly1_2, …, poly1_39，并配合文本标注（Textfield）呈现不同构造步骤或关键点位置，从而形成“开放中间问题”的多解/可变观察。适合中学到大学基础几何课程，用于几何建模、作图思维训练与课堂探究活动。

GeoGebra 命令及参数

  1. Polygon((0, 7), (1, 7), 4) → poly1
  2. Polygon((9, 8), (10, 8), 4) → poly1_{1}
  3. Polygon((4, 9), (5, 9), 4) → poly1_{2}
  4. Polygon((8, 9), (9, 9), 4) → poly1_{3}
  5. Polygon((10, 9), (11, 9), 4) → poly1_{4}
  6. Polygon((3, 8), (4, 8), 4) → poly1_{5}
  7. Polygon((6, 9), (7, 9), 4) → poly1_{6}
  8. Polygon((5, 8), (6, 8), 4) → poly1_{7}
  9. Polygon((1, 8), (2, 8), 4) → poly1_{8}
  10. Polygon((1, 6), (2, 6), 4) → poly1_{9}
  11. Polygon((7, 8), (8, 8), 4) → poly1_{10}
  12. Polygon((11, 8), (12, 8), 4) → poly1_{11}
  13. Polygon((0, 9), (1, 9), 4) → poly1_{12}
  14. Polygon((2, 7), (3, 7), 4) → poly1_{13}
  15. Polygon((2, 9), (3, 9), 4) → poly1_{14}
  16. Polygon((4, 7), (5, 7), 4) → poly1_{15}
  17. Polygon((3, 6), (4, 6), 4) → poly1_{16}
  18. Polygon((6, 7), (7, 7), 4) → poly1_{17}
  19. Polygon((5, 6), (6, 6), 4) → poly1_{18}
  20. Polygon((8, 7), (9, 7), 4) → poly1_{19}
  21. Polygon((7, 6), (8, 6), 4) → poly1_{20}
  22. Polygon((10, 7), (11, 7), 4) → poly1_{21}
  23. Polygon((9, 6), (10, 6), 4) → poly1_{22}
  24. Polygon((0, 5), (1, 5), 4) → poly1_{23}
  25. Polygon((11, 6), (12, 6), 4) → poly1_{24}
  26. Polygon((2, 5), (3, 5), 4) → poly1_{25}
  27. Polygon((1, 4), (2, 4), 4) → poly1_{26}
  28. Polygon((4, 5), (5, 5), 4) → poly1_{27}
  29. Polygon((3, 4), (4, 4), 4) → poly1_{28}
  30. Polygon((6, 5), (7, 5), 4) → poly1_{29}
  31. Polygon((5, 4), (6, 4), 4) → poly1_{30}
  32. Polygon((8, 5), (9, 5), 4) → poly1_{31}
  33. Polygon((7, 4), (8, 4), 4) → poly1_{32}
  34. Polygon((10, 5), (11, 5), 4) → poly1_{33}
  35. Polygon((9, 4), (10, 4), 4) → poly1_{34}
  36. Polygon((0, 3), (1, 3), 4) → poly1_{35}
  37. Polygon((11, 4), (12, 4), 4) → poly1_{36}
  38. Polygon((2, 3), (3, 3), 4) → poly1_{37}
  39. Polygon((1, 2), (2, 2), 4) → poly1_{38}
  40. Polygon((4, 3), (5, 3), 4) → poly1_{39}
  41. Polygon((3, 2), (4, 2), 4) → poly1_{40}
  42. Polygon((6, 3), (7, 3), 4) → poly1_{41}
  43. Polygon((5, 2), (6, 2), 4) → poly1_{42}
  44. Polygon((8, 3), (9, 3), 4) → poly1_{43}
  45. Polygon((7, 2), (8, 2), 4) → poly1_{44}
  46. Polygon((10, 3), (11, 3), 4) → poly1_{45}
  47. Polygon((9, 2), (10, 2), 4) → poly1_{46}
  48. Polygon((0, 1), (1, 1), 4) → poly1_{47}
  49. Polygon((11, 2), (12, 2), 4) → poly1_{48}
  50. Polygon((2, 1), (3, 1), 4) → poly1_{49}
  51. Polygon((1, 0), (2, 0), 4) → poly1_{50}
  52. Polygon((4, 1), (5, 1), 4) → poly1_{51}
  53. Polygon((3, 0), (4, 0), 4) → poly1_{52}
  54. Polygon((6, 1), (7, 1), 4) → poly1_{53}
  55. Polygon((5, 0), (6, 0), 4) → poly1_{54}
  56. Polygon((8, 1), (9, 1), 4) → poly1_{55}
  57. Polygon((7, 0), (8, 0), 4) → poly1_{56}
  58. Polygon((10, 1), (11, 1), 4) → poly1_{57}
  59. Polygon((9, 0), (10, 0), 4) → poly1_{58}
  60. Polygon((11, 0), (12, 0), 4) → poly1_{60}
  61. Textfield(a) → InputBox1

AI 绘图提示语参考

绘制一系列矩形，每个矩形由两个顶点组成，从左到右、从上到下依次排列。首先绘制底部一行矩形，从左到右依次为(0,7)到(1,7)，(2,7)到(3,7)，以此类推，直到(12,7)。然后绘制第二行矩形，从(0,6)到(1,6)，(2,6)到(3,6)，直到(12,6)。继续绘制剩余行，直到(12,1)。
以上提示语为根据原始绘图命令整理而来，仅供参考。