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Wordless Pythagorean Theorem based on Translations

Wordless Pythagorean Theorem based on Translations
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2014-09-03 09:02
TriangleVectorCoordinate GeometryTransformationGeometric Construction
In collections Similarity & Right Triangle Trigonometry Similarity, Right Triangles, and Trigonometry (SRT) Cluster
This example visually demonstrates the Pythagorean Theorem through translations, illustrating the relationship between the sides of a right triangle. Key concepts include the Pythagorean Theorem, vector translations, and geometric constructions. It uses commands like Circle, Line, Point, Intersect, Segment, Distance, OrthogonalLine, Polygon, Vector, Translate. Suitable for high school geometry education and understanding spatial transformations and proofs.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Line(A, B) → a
  3. Point(c) → C
  4. Intersect(c, a) → D, E
  5. Segment(C, E) → b
  6. Segment(C, B) → d
  7. Distance(E, B) → e
  8. OrthogonalLine(C, b) → f
  9. OrthogonalLine(E, b) → g
  10. OrthogonalLine(E, a) → h
  11. OrthogonalLine(B, a) → i
  12. OrthogonalLine(B, f) → j
  13. OrthogonalLine(C, f) → k
  14. Circle(E, B) → p
  15. Circle(C, E) → q
  16. Circle(C, B) → r
  17. Intersect(p, h) → F, G
  18. Intersect(q, f) → H, I
  19. Intersect(r, k) → J, K
  20. OrthogonalLine(F, h) → l
  21. OrthogonalLine(H, g) → m
  22. OrthogonalLine(K, k) → n
  23. Intersect(j, n) → L
  24. Intersect(l, i) → M
  25. Intersect(m, g) → N
  26. Polygon(C, H, N, E) → poly1, c_1, h_1, n_1, e_1
  27. Polygon(C, K, L, B) → poly2, c_2, k_1, l_1, b_1
  28. Polygon(E, F, M, B) → poly3, e_2, f_1, m_1, b_2
  29. Line(M, B) → s
  30. Intersect(s, k_1) → O
  31. Vector(O, E) → u
  32. Translate(B, u) → B'
  33. Translate(L, u) → L'
  34. Translate(O, u) → O'
  35. Vector(B, M) → v
  36. Translate(O, v) → O'_1
  37. Translate(K, v) → K'
  38. Translate(C, v) → C'
  39. Translate(B, v) → B'_1
  40. Intersect(c_2, s) → R
  41. Polygon(R, C, B) → poly4, b_3, r_1, c_3
  42. Polygon(B, L, K, R) → poly5, b_4, l_2, k_2, r_2
  43. OrthogonalLine(N, s) → t
  44. Intersect(t, c_1) → S
  45. OrthogonalLine(S, t) → a_1
  46. Intersect(b, e_1) → T
  47. Intersect(a_1, b) → U
  48. Vector(B, F) → w
  49. Translate(B, w) → B'_2
  50. Translate(L, w) → L'_1
  51. Translate(K, w) → K'_1
  52. Translate(R, w) → R'
  53. Polygon(U, C, S) → poly6, s_1, u_1, c_4
  54. Vector(U, R') → z
  55. Point(poly6) → V
  56. Polygon(N, S, U, E) → poly7, n_2, s_2, u_2, e_3
  57. Intersect(a_1, c_1) → W
  58. Polygon(H, N, W) → poly8, w_1, h_2, n_3
  59. Translate(B, w) → B'_3
  60. Translate(L, w) → L'_2
  61. Translate(K, w) → K'_2
  62. Translate(R, w) → R'_1
  63. Polygon(B'_3, L'_2, K'_2, R'_1) → poly5', b'_3, l'_2, k'_2, r'_1
  64. Translate(U, z) → U'
  65. Translate(C, z) → C'_1
  66. Translate(S, z) → S'
  67. Polygon(U', C'_1, S') → poly6', s', u', c'_1
  68. Vector(N, B'_3) → v_1
  69. Translate(H, v_1) → H'
  70. Translate(N, v_1) → N'
  71. Translate(W, v_1) → W'
  72. Polygon(H', N', W') → poly8', w', h', n'
  73. Translate(R, v) → R'_2
  74. Translate(C, v) → C'_2
  75. Translate(B, v) → B'_4
  76. Polygon(R'_2, C'_2, B'_4) → poly4', b'_4, r'_2, c'_2
  77. Vector(S', L'_2) → z_1
  78. Translate(N, z_1) → N'_1
  79. Translate(S, z_1) → S'_1
  80. Translate(U, z_1) → U'_1
  81. Translate(E, z_1) → E'
  82. Polygon(N'_1, S'_1, U'_1, E') → poly7', n'_1, s'_1, u'_1, e'

AI Drawing Prompt Reference

Draw a right triangle to demonstrate the Pythagorean theorem through translations. Start by sketching the legs and hypotenuse, then draw perpendicular lines on the legs and hypotenuse to find intersection points. Construct new shapes using circles and these points, showing translation relationships. Finally, form multiple closed figures by translating and connecting points, visually representing the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.