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Triangle Area

Triangle Area
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2016-12-22 04:03
TriangleCoordinate GeometryGeometric Construction
This example demonstrates calculating the area of a triangle through rotation and constructing polygons. Key mathematical concepts include area calculation, properties of equilateral triangles, rotation, and translation. It uses commands like Segment, Polygon, Rotate, Line, and Midpoint in GeoGebra. Suitable for middle school and above students to learn geometric construction and area calculation.

GeoGebra Commands & Parameters

  1. Segment(A, Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]]) → j
  2. Polygon(A, B, C) → poly1, c, a, b
  3. Polygon(B, Midpoint[A, B], Midpoint[A, C], C) → poly2, b_1, d, e, c_1
  4. Rotate(Polygon[A, Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], Midpoint[A, B]], α, Midpoint[A, B]) → poly3
  5. Rotate(Polygon[A, Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], Midpoint[A, C]], (-α), Midpoint[A, C]) → poly4
  6. Segment(A, Intersect[OrthogonalLine[A, Line[B, C]], Line[Midpoint[A, B], Midpoint[A, C]]]) → m
  7. Segment(Intersect[OrthogonalLine[A, Line[B, C]], Line[Midpoint[A, B], Midpoint[A, C]]], Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]]) → n
  8. Segment(A, Midpoint[A, B]) → p
  9. Segment(A, Midpoint[A, C]) → q
  10. Rotate(Polygon[A, Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], C], γ, Midpoint[A, C]) → poly5
  11. Rotate(Polygon[A, Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], B], (-γ), Midpoint[A, B]) → poly6
  12. Angle(Midpoint[A, C], Rotate[Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], (-α), Midpoint[A, C]], Rotate[A, (-α), Midpoint[A, C]]) → ι
  13. Angle(Rotate[A, α, Midpoint[A, B]], Rotate[Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], α, Midpoint[A, B]], Midpoint[A, B]) → η
  14. Angle(Rotate[A, γ, Midpoint[A, C]], Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], γ, Midpoint[A, C]], Rotate[C, γ, Midpoint[A, C]]) → θ
  15. Angle(Rotate[B, (-γ), Midpoint[A, B]], Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], (-γ), Midpoint[A, B]], Rotate[A, (-γ), Midpoint[A, B]]) → ε
  16. Angle(C, B, Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], (-γ), Midpoint[A, B]]) → δ_1
  17. Angle(Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], γ, Midpoint[A, C]], C, B) → γ_1
  18. Angle(Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], γ, Midpoint[A, C]], Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], (-γ), Midpoint[A, B]], B) → ζ
  19. Angle(Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], (-γ), Midpoint[A, B]], Rotate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], γ, Midpoint[A, C]], C) → κ
  20. Rotate(Polygon[Midpoint[A, C], Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], C], β, Midpoint[A, C]) → poly7
  21. Rotate(Polygon[Midpoint[A, B], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]], B], (-β), Midpoint[A, B]) → poly8
  22. Polygon(Midpoint[A, B], A, Midpoint[A, C], Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]]) → poly9, l, s, a_1, d_1, e_1
  23. Angle(Midpoint[A, C], Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]]) → σ
  24. Angle(Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]], Midpoint[A, B]) → τ
  25. Angle(Rotate[C, β, Midpoint[A, C]], Rotate[Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], β, Midpoint[A, C]], Midpoint[A, C]) → υ
  26. Angle(Midpoint[A, B], Rotate[Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]], (-β), Midpoint[A, B]], Rotate[B, (-β), Midpoint[A, B]]) → ϕ
  27. Segment(Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], C) → f_1
  28. Segment(C, Midpoint[A, C]) → g_1
  29. Segment(Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]]) → h_1
  30. Segment(B, Midpoint[A, B]) → i_1
  31. Segment(B, Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]]) → j_1
  32. Segment(Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]]) → k_1
  33. Angle(Rotate[Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], (-α), Midpoint[A, C]], C, B) → χ
  34. Angle(C, B, Rotate[Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], α, Midpoint[A, B]]) → ψ
  35. Angle(B, Midpoint[A, B], Rotate[Midpoint[A, B], (-α), Midpoint[A, C]]) → ρ
  36. Angle(B, A, Rotate[Midpoint[B, C], β, Midpoint[A, C]]) → ω
  37. Angle(C, A, Rotate[Midpoint[B, C], β, Midpoint[A, B]]) → α_1
  38. Polygon(Rotate[A, δ, Midpoint[A, C]], Rotate[B, δ, Midpoint[A, C]], Rotate[C, δ, Midpoint[A, C]]) → poly1', r_1, s_1, t_1
  39. Polygon(B, Dilate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], g, A], C) → poly10, l_1, m_1, n_1
  40. Segment(Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], Dilate[Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], g, A]) → p_1
  41. Line(B, C) → q_1
  42. Angle(Rotate[B, δ, Midpoint[A, C]], A, Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]]) → λ
  43. Polygon(A, Intersect[OrthogonalLine[A, Line[B, C]], Line[Midpoint[A, B], Midpoint[A, C]]], Midpoint[A, B]) → poly11, a_2, a_3, b_2
  44. Polygon(A, Intersect[OrthogonalLine[A, Line[B, C]], Line[Midpoint[A, B], Midpoint[A, C]]], Midpoint[A, C]) → poly12, c_2, a_4, d_2
  45. Polygon(Midpoint[A, C], Intersect[OrthogonalLine[Midpoint[A, C], Line[B, C]], Line[B, C]], C) → poly13, c_3, e_2, f_2
  46. Polygon(Midpoint[A, B], Intersect[OrthogonalLine[Midpoint[A, B], Line[B, C]], Line[B, C]], B) → poly14, b_3, h_2, i_2
  47. Rotate(A, (-γ), Midpoint[A, B]) → A'
  48. Rotate(Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], (-γ), Midpoint[A, B]) → D
  49. Rotate(B, (-γ), Midpoint[A, B]) → B'
  50. Segment(A', D) → j_2
  51. Segment(D, B') → k_2
  52. Segment(B', A') → l_2
  53. Rotate(A, γ, Midpoint[A, C]) → A'_1
  54. Rotate(C, γ, Midpoint[A, C]) → C'
  55. Rotate(Intersect[OrthogonalLine[A, Line[B, C]], Line[B, C]], γ, Midpoint[A, C]) → E
  56. Segment(C', E) → m_2
  57. Segment(A'_1, C') → n_2
  58. Segment(E, A'_1) → p_2
  59. Rotate(A, α, Midpoint[A, B]) → A'_2
  60. Midpoint(A, B) → F
  61. Rotate(Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], α, Midpoint[A, B]) → G
  62. Segment(A'_2, G) → f
  63. Segment(G, F) → h
  64. Segment(F, A'_2) → i
  65. Rotate(A, (-α), Midpoint[A, C]) → A'_3
  66. Midpoint(A, C) → H
  67. Rotate(Intersect[OrthogonalLine[A, Line[Midpoint[A, B], Midpoint[A, C]]], Line[Midpoint[A, B], Midpoint[A, C]]], (-α), Midpoint[A, C]) → I
  68. Segment(I, A'_3) → k
  69. Segment(A'_3, H) → r
  70. Segment(H, I) → t