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Pythagoras' Theorem - A Visual Demonstration

Pythagoras' Theorem - A Visual Demonstration
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2014-06-09 16:47
TrianglePolygonVectorTransformationGeometric Construction
① This example provides a visual demonstration of Pythagoras' Theorem, showing that in a right triangle, the sum of the areas of the squares on the two legs equals the area of the square on the hypotenuse. ② The core mathematical concept is the Pythagorean theorem, involving right triangles, square areas, and geometric transformations. ③ Construction details: using GeoGebra commands such as Intersect, Point, Polygon, Midpoint, Line, OrthogonalLine, Vector, and Translate, the example constructs a right triangle and squares on each side, then uses translations and vectors to illustrate the area relationship; objects include points A, B, C, polygons poly1 through poly5, and auxiliary lines and vectors. ④ Suitable for middle school geometry classes to intuitively understand the theorem, and for math labs or exploratory activities.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Point(yAxis) → C
  4. Polygon(A, C, B) → poly1, b, a, c
  5. Polygon(A, C, 4) → poly2, a_1, c_1, d, e, D, E
  6. Polygon(C, B, 4) → poly3, c_2, b_1, f, g, F, G
  7. Polygon(B, A, 4) → poly4, b_2, a_2, h, i, H, I
  8. Midpoint(A, I) → J
  9. Line(J, a) → j
  10. OrthogonalLine(J, a) → k
  11. Intersect(j, a_2) → K
  12. Intersect(j, i) → L
  13. Intersect(k, c) → M
  14. Intersect(k, h) → N
  15. Polygon(A, K, J, M) → poly5, a_3, k_1, j_1, m
  16. Polygon(K, H, N, J) → poly6, k_2, h_1, n, j_2
  17. Polygon(I, N, J, L) → poly7, i_1, n_1, j_3, l
  18. Polygon(L, B, M, J) → poly8, l_1, b_3, m_1, j_4
  19. Vector(J, G) → u
  20. Vector(J, C) → v
  21. Vector(J, B) → w
  22. Vector(J, F) → z
  23. Translate(I, u) → I'
  24. Translate(N, u) → N'
  25. Translate(J, u) → J'
  26. Translate(L, u) → L'
  27. Translate(L, v) → L'_1
  28. Translate(B, v) → B'
  29. Translate(M, v) → M'
  30. Translate(J, v) → J'_1
  31. Translate(K, z) → K'
  32. Translate(H, z) → H'
  33. Translate(N, z) → N'_1
  34. Translate(J, z) → J'_2
  35. Translate(A, p) → A'
  36. Translate(K, p) → K'_1
  37. Translate(J, p) → J'_3
  38. Translate(M, p) → M'_1
  39. Polygon(A', K'_1, J'_3, M'_1) → poly5', a', k'_1, j'_3, m'_1
  40. Translate(A', w) → A''
  41. Translate(K'_1, w) → O
  42. Translate(J'_3, w) → P
  43. Translate(M'_1, w) → Q
  44. Translate(K, rightpiece) → K'_2
  45. Translate(H, rightpiece) → H'_1
  46. Translate(N, rightpiece) → N'_2
  47. Translate(J, rightpiece) → J'_4
  48. Polygon(K'_2, H'_1, N'_2, J'_4) → left, k'_2, h'_1, n'_2, j'_4
  49. Translate(I, toppiece) → I'_1
  50. Translate(N, toppiece) → N'_3
  51. Translate(J, toppiece) → J'_5
  52. Translate(L, toppiece) → L'_2
  53. Polygon(I'_1, N'_3, J'_5, L'_2) → bottom, i'_1, n'_3, j'_5, l'_2
  54. Translate(L, leftpiece) → L'_3
  55. Translate(B, leftpiece) → B'_1
  56. Translate(M, leftpiece) → M'_2
  57. Translate(J, leftpiece) → J'_6
  58. Polygon(L'_3, B'_1, M'_2, J'_6) → right, l'_3, b'_1, m'_2, j'_6
  59. Translate(A', bottompiece) → A''_1
  60. Translate(K'_1, bottompiece) → R
  61. Translate(J'_3, bottompiece) → S
  62. Translate(M'_1, bottompiece) → T
  63. Polygon(A''_1, R, S, T) → top, a''_1, r, s, t
  64. Intersect(f, b_1) → U
  65. Midpoint(C, U) → V
  66. Intersect(d, e) → W
  67. Midpoint(C, W) → Z
  68. Vector(Z, V) → q
  69. Translate(poly2, otherpiece) → smaller

AI Drawing Prompt Reference

Draw a right triangle ABC, then construct squares on each side. Use translations and rotations to rearrange these squares to show that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
The prompt above was compiled from the original drawing commands and is for reference only.