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Pythagorean Theorem Demonstration by PopiArdv (2)

Pythagorean Theorem Demonstration by PopiArdv (2)
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2013-01-14 06:59
TriangleCoordinate GeometryGeometric Construction
This example visually demonstrates the Pythagorean theorem in right-angled triangles, showcasing the relationship between the squares of the side lengths. Key math concepts include the Pythagorean theorem, right triangles, and coordinate geometry. It utilizes commands and objects like Point, Polygon, Circle, Intersect, and Midpoint from GeoGebra. Suitable for middle school students and above for learning and teaching purposes.

GeoGebra Commands & Parameters

  1. Semicircle(Α, Β) → c
  2. Point(c) → Γ
  3. Polygon(Γ, Α, Β) → πολύγ1, β, γ, α
  4. Polygon(Α, Γ, 4) → πολύγ2, a, b, d, e, Δ, Ε
  5. CircleArc(Α, Γ, Ε) → f
  6. Rotate(Γ, δ, Α) → Γ'
  7. Angle(Γ, Α, Γ') → ε
  8. Circle(Ε, ζ) → g
  9. Intersect(g, d, 1) → Ζ
  10. Polygon(Γ, Β, 4) → πολύγ3, h, i, j, k, Η, Θ
  11. CircleArc(Β, Η, Γ) → p
  12. Rotate(Γ, -(δ - 150°), Β) → Γ'_1
  13. Angle(Γ'_1, Β, Γ) → η
  14. Circle(Η, l) → q
  15. Intersect(q, j) → Ι, Κ
  16. Midpoint(γ) → Λ
  17. Rotate(Γ, θ, Λ) → Γ'_2
  18. Angle(Γ, Λ, Γ'_2) → ι
  19. Mirror(Γ, Λ) → Γ'_3
  20. Circle(Ε, α) → r
  21. Ray(Ε, Δ) → m
  22. Intersect(r, m, 1) → Μ
  23. Circle(Α, n) → s
  24. OrthogonalLine(Α, γ) → t
  25. Ray(Ε, Δ) → a_1
  26. Ray(Η, Θ) → b_1
  27. Intersect(a_1, b_1) → Ν
  28. Segment(Μ, Α) → c_1
  29. Angle(Γ, Λ, Γ'_2) → κ
  30. Rotate(Α, κ, Λ) → Α'
  31. Angle(Α, Λ, Α') → λ
  32. Mirror(Α', Λ) → Α''
  33. Vector(Α, Β) → u
  34. Segment(Γ, Ν) → e_1
  35. Segment(Α, Γ') → f_1
  36. OrthogonalLine(Γ', f_1) → g_1
  37. Circle(Γ', α) → h_1
  38. Intersect(h_1, g_1) → Π, Ρ
  39. Polygon(Π, Γ', Α) → πολύγ8, α_2, π, γ'
  40. Polygon(Μ, Ε, Α) → πολύγ9, α_3, μ_1, ε_1
  41. Vector(Ε, Α) → v
  42. Translate(Ζ, v) → Ζ'
  43. Circle(Ζ, α) → k_1
  44. Intersect(k_1, a_1, 1) → Σ
  45. Polygon(Σ, Ζ, Ζ') → πολύγ10, ζ', σ, ζ_1
  46. Polygon(Ν, Γ, Δ) → πολύγ11, δ_1, ν_1, γ_1
  47. Segment(Γ'_1, Β) → i_1
  48. OrthogonalLine(Γ'_1, i_1) → j_1
  49. Circle(Γ'_1, β) → p_1
  50. Intersect(p_1, j_1, 1) → Τ
  51. Polygon(Τ, Γ'_1, Β) → πολύγ12, β_2, τ, γ'_1
  52. Intersect(k_1, a_1, 2) → Υ
  53. Polygon(Υ, Ζ, Ζ') → πολύγ14, ζ'_1, υ, ζ_2
  54. Vector(Η, Β) → w
  55. Translate(Κ, w) → Κ'
  56. Circle(Κ, β) → q_1
  57. Intersect(j, b_1) → Φ
  58. Intersect(j, b_1) → Χ
  59. Intersect(q_1, b_1) → Ψ, Ω
  60. Polygon(Ν, Θ, Γ) → πολύγ16, γ_2, ν_2, θ_1
  61. Rotate(Γ, (δ - (180 + 2 * w_1)°) * 3, Λ) → Γ'_4
  62. Angle(Γ, Λ, Γ'_4) → ρ
  63. Polygon(Ω, Κ, Κ') → πολύγ15, κ', ω, κ_1
  64. Rotate(Α, 3 * (δ - (180 + 2 * w_1)°), Λ) → Α'_1
  65. Angle(Α, Λ, Α'_1) → ϕ
  66. Mirror(Α'_1, Λ) → Α_1
  67. Polygon(Α_1, Γ'_4, Α'_1) → πολύγ17, α'_1, α_4, γ'_4
  68. Segment(Ε, Ν) → l_1
  69. Segment(Ν, Η) → m_1
  70. Translate(Ι, w) → Ι'
  71. Circle(Ι, β) → r_1
  72. Intersect(r_1, b_1) → Β_1, Γ_1
  73. Polygon(Ι', Ι, Γ_1) → πολύγ7, m_2, n_2, p_2
  74. Ray(Η, Β) → n_1
  75. Ray(Ε, Α) → s_1
  76. Intersect(n_1, s_1) → Δ_1
  77. Segment(Δ_1, Η) → t_1
  78. Segment(Ε, Δ_1) → a_2
  79. Vector(Α, Γ) → z
  80. Translate(Ξ, u) → Ξ'
  81. Translate(Ξ, z) → Ξ'_1
  82. Polygon(Ξ'_1, Ξ, Ξ') → πολύγ4, ξ', ξ'_1, ξ
  83. Polygon(Ξ', Ξ, 4) → πολύγ5, d_1, b_2, c_2, d_2, Ο, Ε_1
  84. Rotate(Ξ'_1, θ, Ξ) → Ζ_1
  85. Angle(Ξ'_1, Ξ, Ζ_1) → μ
  86. Segment(Ξ, Ζ_1) → e_2
  87. OrthogonalLine(Ζ_1, e_2) → f_2
  88. Circle(Ζ_1, α) → g_2
  89. Intersect(g_2, f_2, 1) → Θ_1
  90. Polygon(Θ_1, Ζ_1, Ξ) → πολύγ6, ξ_1, θ_2, ζ_3
  91. Rotate(Ξ'_1, -(θ - 90°), Ξ') → Ι_1
  92. Angle(Ι_1, Ξ', Ξ'_1) → ν
  93. Segment(Ι_1, Ξ') → h_2
  94. OrthogonalLine(Ι_1, h_2) → i_2
  95. Circle(Ι_1, β) → k_2
  96. Intersect(k_2, i_2, 1) → Κ_1
  97. Polygon(Ξ', Ι_1, Κ_1) → πολύγ13, κ_2, ξ'_2, ι_1
  98. Vector(Ο, Ξ) → u_1
  99. Translate(Ξ, u_1) → Ξ'_2
  100. Translate(Ξ', u_1) → Ξ''
  101. Rotate(Ξ'_1, 90*°, Ξ) → Η_1
  102. Angle(Ξ'_1, Ξ, Η_1) → ο
  103. Rotate(Ξ'_1, 270*°, Ξ') → Λ_1
  104. Angle(Λ_1, Ξ', Ξ'_1) → χ
  105. Polygon(Ξ'_2, Ξ, Η_1) → πολύγ18, η_1, ξ'_3, ξ_2
  106. Polygon(Ξ'', Λ_1, Ξ') → πολύγ19, ξ'_4, ξ'', λ_1
  107. Circle(Ο, o_1) → q_2
  108. Ray(Ο, Ξ) → j_2
  109. Intersect(q_2, j_2) → Μ_1, Χ_1
  110. Translate(Μ_1, u_1) → Ν_1
  111. Polygon(Ν_1, Μ_1, 4) → πολύγ20, l_2, r_2, s_2, t_2, Ο_1, Π_1
  112. Circle(Ξ, o) → c_3
  113. Intersect(c_3, j_2) → Ρ_1, Σ_1
  114. Translate(Σ_1, u) → Τ_1
  115. Vector(Ξ, Ξ'_1) → v_1
  116. Translate(Σ_1, v_1) → Υ_1
  117. Polygon(Υ_1, Σ_1, Τ_1) → πολύγ21, τ_1, υ_1, σ_1
  118. Translate(Ξ', u_1) → Ξ''_1
  119. Translate(Ξ, u_1) → Ξ'_3
  120. Translate(Ξ'_3, v_1) → Φ_1
  121. Polygon(Φ_1, Ξ'_3, Ξ''_1) → πολύγ22, ξ''_1, φ_1, ξ'_5
  122. Translate(Χ_1, u) → Ψ_1
  123. Polygon(Χ_1, Ψ_1, 4) → πολύγ23, a_3, b_3, d_3, e_3, Ω_1, Α_2
  124. Midpoint(Ξ, Ξ') → Β_2
  125. Rotate(Ξ'_1, n, Β_2) → Γ_2
  126. Angle(Ξ'_1, Β_2, Γ_2) → ψ
  127. Rotate(Ξ, n, Β_2) → Ξ'_4
  128. Angle(Ξ, Β_2, Ξ'_4) → α_1
  129. Mirror(Ξ'_4, Β_2) → Δ_2
  130. Polygon(Δ_2, Γ_2, Ξ'_4) → πολύγ24, ξ'_6, δ_2, γ_3
  131. Translate(Ξ', v_1) → Ξ''_2
  132. Mirror(Ξ''_2, Ξ') → Ε_2
  133. Polygon(Ξ, Ξ', Ε_2) → πολύγ25, ε_2, ξ_3, ξ'_7
  134. Segment(Φ_1, Η_1) → f_3
  135. Segment(Η_1, Ε_2) → g_3
  136. Segment(Ε_2, Λ_1) → h_3
  137. Segment(Λ_1, Φ_1) → i_3
  138. Segment(Γ, Δ) → j_3

AI Drawing Prompt Reference

Draw a right triangle and construct squares on its two legs, visually demonstrating the Pythagorean theorem where the area of the hypotenuse's square equals the sum of the legs' squares. Use circles, rotations, and coordinate geometry to create auxiliary shapes, labeling key points and segments, to illustrate the theorem's proof.
The prompt above was compiled from the original drawing commands and is for reference only.