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勾股定理应用：直角三角形斜边计算

勾股定理应用：直角三角形斜边计算
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2024-09-26 21:25
测试改名0角几何作图
所属专题 Satz des Pythagoras - Einführung
本范例通过互动练习帮助学生掌握勾股定理的实际应用。核心内容包含：①根据已知两直角边（a=5cm, b=12cm或a=15cm, b=8cm）计算斜边长度；②理解勾股定理多种表达形式（a²+b²=c², e²=f²+g²等）；③探讨不同符号表示的等价性。GeoGeba构造使用了Textfield输入框、Point点、Segment线段、Polygon多边形、Angle角度、Distance距离、Rotate旋转等命令绘制直角三角形并动态显示计算过程。适合7-8年级学生使用，可用于课堂练习或自主复习，培养学生正确书写和运用勾股定理的能力。

GeoGebra 命令及参数

  1. Textfield(input12) → Eingabefeld12
  2. SelectedIndex(DropDown21) → input21
  3. Element({4, 5, 6, 6.1, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19}, step2) → step
  4. If(step ≟ 16, (11.4, -3.4), If[step ≥ 17, A1Ende, A1Start]) → A1
  5. SelectedIndex(DropDown3) → input3
  6. SelectedIndex(DropDown1) → input1
  7. SelectedIndex(DropDown5) → input5
  8. SelectedIndex(DropDown6) → input6
  9. SelectedIndex(DropDown7) → input7
  10. Vector(C1, A1) → v2_{1}
  11. Vector(C1, B1) → v1_{1}
  12. Sequence(O1 + (m * (v2_{1} + (0.6, 0))), m, 0, 3) → listO
  13. Point(Segment[listO(4), O2]) → C4
  14. Textfield(input9) → Eingabefeld9
  15. Point(Segment[listO(3), O3]) → C3
  16. Point(Segment[listO(2), O4]) → C2
  17. Point(Segment[B1, C1]) → S1
  18. Point(Segment[A1, C1]) → S2
  19. Textfield(input2) → Eingabefeld2
  20. Point(Line[V1, xAxis]) → V2
  21. Segment(B1, A1) → s3
  22. Segment(B1, S1) → s1
  23. Segment(A1, S2) → s2
  24. Angle(Polygon[C1, A1, B1]) → wa3, wa2, wa1
  25. Polygon(A2, B2, C2) → d2, t3, t1, t2
  26. Polygon(B3, C3, A3) → d3, u1, u2, u3
  27. Polygon(A4, B4, C4) → d4, v3, v1, v2
  28. Angle(d2) → wb2, wb1, wb3
  29. Angle(d3) → wc1, wc3, wc2
  30. Angle(d4) → wd2, wd1, wd3
  31. Zip(Midpoint[A, B] + (0.2 * Vector[C, Midpoint[A, B]]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM3
  32. Zip(Midpoint[B, C] + (0.15 * Vector[A, C]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM1
  33. Zip(Midpoint[A, C] + (0.09 * Vector[B, C]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM2
  34. Zip(B + (0.19 * Vector[B, C + (1 / 2 * (A - C))]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMB
  35. Angle(A1, B1, B2) → wa4
  36. Angle(A2, B2, B3) → wb4
  37. Angle(A3, B3, B4) → wc4
  38. Angle(A4, B4, B1) → wd4
  39. Point(Line[L1, xAxis]) → L2
  40. Zip(A + (0.18 * Vector[C, B]) + (0.08 * Vector[C, A]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMD
  41. Point(Line[L3, xAxis]) → L4
  42. Distance(L3, L4) → lupesize2
  43. Rotate(R1a, wa1, Q1) → R1b
  44. Angle(R1a, Q1, R1b) → va1
  45. Rotate(R1a, wa1 + wa3, Q1) → R1c
  46. Rotate(R1c, wa2, Q1) → R1d
  47. Angle(R1c, Q1, R1d) → va2
  48. Angle(R1b, Q1, R1c) → va3
  49. Rotate(R2a, wa1, Q2) → R2b
  50. Angle(R2a, Q2, R2b) → vb1
  51. Rotate(R2b, wa3, Q2) → R2c
  52. Rotate(R2c, wa2, Q2) → R2d
  53. Angle(R2b, Q2, R2c) → vb3
  54. Angle(R2c, Q2, R2d) → vb2
  55. Rotate(R3a, (-wa2), Q3) → R3b
  56. Rotate(R3b, (-wa3), Q3) → R3c
  57. Rotate(R3c, (-wa1), Q3) → R3d
  58. Angle(R3d, Q3, R3c) → vc1
  59. Angle(R3b, Q3, R3a) → vc2
  60. Angle(R3c, Q3, R3b) → vc3
  61. Rotate(R4a, wa1, Q4) → R4b
  62. Rotate(R4b, wa3, Q4) → R4c
  63. Rotate(R4c, wa2, Q4) → R4d
  64. Angle(R4a, Q4, R4b) → vd1
  65. Angle(R4b, Q4, R4c) → vd3
  66. Angle(R4c, Q4, R4d) → vd2
  67. Textfield(input4) → Eingabefeld4
  68. Polygon(A1, B1, C1) → d1, c1, a1, b1
  69. Textfield(input14) → Eingabefeld14
  70. Textfield(input15) → Eingabefeld15
  71. Textfield(input16) → Eingabefeld16
  72. Textfield(input17) → Eingabefeld17
  73. Textfield(input18) → Eingabefeld18
  74. Textfield(input19) → Eingabefeld19
  75. LaTeX(step, true, true) → Text12
  76. Polygon(DFlink3, DFlink1, DFlink2) → dflink, dflink3, dflink1, dflink2
  77. Angle(dflink) → wflink1, wflink2, wflink3
  78. Textfield(input20) → Eingabefeld20
  79. CountIf(m ≟ true, m, {selected1, selected2, selected3, selected4}) → numselected
  80. Textfield(input10) → Eingabefeld10
  81. Textfield(input11) → Eingabefeld11
  82. Textfield(input13) → Eingabefeld13
  83. If(path1, Pos0 + (1.3, -1.9), Pos0 + (1.3, -3.3)) → Pos9
  84. If(path1, Pos0 + (3.45, -2.02), Pos0 + (3.45, -3.6)) → Pos10
  85. If(path1, Pos0 + (5.75, -2.02), Pos0 + (5.75, -3.6)) → Pos11
  86. Element(Join[{Sequence[Posi0 + (m * (0, -1.22)) + (1.1, -1), m, 0, 2], {Posi0 + (2.4, -5.2)}}], If[correct18, 4, If[correct16 ∧ correct17, 3, If[correct14 ∧ correct15, 2, 1]]]) → PosiSol
  87. Angle(A1, B1, B2) → wa4punkt
  88. Angle(A2, B2, B3) → wb4punkt
  89. Angle(A3, B3, B4) → wc4punkt
  90. Angle(A4, B4, B1) → wd4punkt
  91. Zip(A + (0.25 * Vector[A, C + (2 / 5 * (B - C))]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMA
  92. If(path1, Pos0 + (4.9, -3.3), Pos0 + (3.7, -1.9)) → Pos12
  93. Element(Sequence[Pos0 + (2, -3.1) + (m * (0, -1.3)), m, 0, 2], If[path1, If[correct12, 3, If[correct9 ∧ correct10 ∧ correct11, 2, 1]], If[correct9 ∧ correct10 ∧ correct11, 3, If[correct12, 2, 1]]]) → PosSol4
  94. If(path1, If[correct12, 3, If[correct9 ∧ correct10 ∧ correct11, 2, 1]], If[correct9 ∧ correct10 ∧ correct11, 3, If[correct12, 2, 1]]) → solPos
  95. Point(Line[Py1, xAxis]) → Py2

AI 绘图提示语参考

绘制一个右三角形的教学几何图：先给定两条互相垂直的边作为直角三角形的两条直角边，并画出对应的斜边，保留顶点与边的连接关系；在三角形内部标注由三条边形成的三个角度，角度分别对应各顶点处的夹角；随后再生成围绕三角形相关边的辅助三角形/多边形，保证它们与原三角形在同一组顶点位置上连接；最后用旋转方式从某个顶点出发构造一处角的对应位置，并再次标注旋转后形成的角度变化，用于解释不同记号下角度或关系的等价性与图形对应。
以上提示语为根据原始绘图命令整理而来，仅供参考。