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Cosine Theorem Explorer: Dynamic Links Between Sides and Angles

Cosine Theorem Explorer: Dynamic Links Between Sides and Angles
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2021-01-17 00:36
TrianglePolygonAngleTrigonometryGeometric Construction
This GeoGebra example dynamically demonstrates the cosine theorem by constructing a triangle and showing how the three side lengths relate to the included angle. It focuses on core ideas from trigonometry—especially the cosine function and its geometric meaning in triangles. Key concepts include translating an angle into a side-length relationship and observing how those quantities change when parameters are varied. Construction uses GeoGebra commands such as Polygon to form the triangle, Angle to define/measure the included angle, Line and OrthogonalLine for auxiliary and perpendicular constructions, Point and Intersect to locate critical vertices and intersection points, and Random together with parameter lists (e.g., listAngle, listSide) to generate interactive configurations. Text/LaTeX objects present computed values and the underlying formula, while buttons/controls manage the animation and workflow. Suitable for secondary school learners through early university students, and for teaching or self-study of trigonometric geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Angle(B, A, C) → α
  3. Angle(C, B, A) → β
  4. Angle(A, C, B) → γ
  5. Random(1, 3) → randSide
  6. Random(1, 3) → randAngle
  7. LaTeX(a, true, true) → text1
  8. LaTeX(b, true, true) → text2
  9. LaTeX(c, true, true) → text3
  10. LaTeX(α, true, true) → text4
  11. LaTeX(β, true, true) → text5
  12. LaTeX(γ, true, true) → text6
  13. Line(I, J) → p
  14. OrthogonalLine(Q, p) → q
  15. OrthogonalLine(R, p) → r
  16. Line(H, I) → s
  17. Point(s) → S
  18. OrthogonalLine(S, s) → t
  19. Point(s) → T
  20. OrthogonalLine(T, s) → f_1
  21. Point(s) → U
  22. OrthogonalLine(U, s) → g_1
  23. Point(s) → V
  24. OrthogonalLine(V, s) → h_1
  25. Point(s) → W
  26. OrthogonalLine(W, s) → i_1
  27. Point(s) → B_1
  28. OrthogonalLine(B_1, s) → j_1
  29. Point(s) → C_1
  30. OrthogonalLine(C_1, s) → k_1
  31. Point(s) → D_1
  32. OrthogonalLine(D_1, s) → l_1
  33. Line(J, K) → m_1
  34. Intersect(q, t) → E_1
  35. Intersect(r, t) → F_1
  36. Intersect(t, m_1) → G_1
  37. Intersect(q, f_1) → H_1
  38. Intersect(q, g_1) → I_1
  39. Intersect(q, h_1) → J_1
  40. Intersect(q, i_1) → K_1
  41. Intersect(q, j_1) → L_1
  42. Intersect(q, k_1) → M_1
  43. Intersect(q, l_1) → N_1
  44. Intersect(p, q) → O_1
  45. Intersect(r, f_1) → P_1
  46. Intersect(r, g_1) → Q_1
  47. Intersect(r, h_1) → R_1
  48. Intersect(r, i_1) → S_1
  49. Intersect(r, j_1) → T_1
  50. Intersect(r, k_1) → U_1
  51. Intersect(r, l_1) → V_1
  52. Intersect(f_1, m_1) → W_1
  53. Intersect(g_1, m_1) → Z_1
  54. Intersect(h_1, m_1) → A_2
  55. Intersect(i_1, m_1) → B_2
  56. Intersect(j_1, m_1) → C_2
  57. Intersect(k_1, m_1) → D_2
  58. Intersect(l_1, m_1) → E_2
  59. Intersect(p, r) → F_2
  60. Polygon(H, S, E_1, Q) → squareA_{1}, h_2, s_1, e_1, q_1
  61. Polygon(H_1, I_1, Q_1, P_1) → squareA_{2}, n_1, p_1, r_1, t_1
  62. Polygon(P_1, Q_1, Z_1, W_1) → squareA_{3}, a_1, b_1, z_1, w_1
  63. Polygon(T, U, I_1, H_1) → squareB_{1}, t_2, u, c_1, d_1
  64. Polygon(B_1, C_1, M_1, L_1) → squareB_{2}, f_2, g_2, i_2, j_2
  65. Polygon(Q, E_1, F_1, R) → squareB_{3}, q_2, k_2, l_2, r_2
  66. Polygon(R_1, S_1, B_2, A_2) → squareB_{4}, m_2, n_2, b_2, a_2
  67. Polygon(L_1, M_1, U_1, T_1) → squareA_{4}, p_2, s_2, u_1, c_2
  68. Polygon(V, W, K_1, J_1) → squareC_{1}, v, w, d_2, e_2
  69. Polygon(C_1, D_1, N_1, M_1) → squareC_{2}, f_3, g_3, h_3, i_3
  70. Polygon(J_1, K_1, S_1, R_1) → squareC_{3}, j_3, k_3, l_3, m_3
  71. Polygon(R, F_1, G_1, K) → squareC_{4}, r_3, n_3, p_3, k_4
  72. Polygon(D_1, I, O_1, N_1) → squareAlpha, q_3, i_4, o_1, s_3
  73. Polygon(N_1, O_1, F_2, V_1) → squareBeta, t_3, a_3, b_3, v_1
  74. Polygon(V_1, F_2, J, E_2) → squareGamma, c_3, d_3, j_4, e_3
  75. Polygon(T_1, U_1, D_2, C_2) → squareA, f_4, g_4, h_4, l_4
  76. Polygon(U_1, V_1, E_2, D_2) → squareB, m_4, n_4, p_4, q_4
  77. Polygon(M_1, N_1, V_1, U_1) → squareC, r_4, s_4, t_4, a_4

AI Drawing Prompt Reference

Draw a triangle with three labeled vertices and mark the three interior angles: form angle α at vertex A (between the two sides meeting at A), angle β at vertex B, and angle γ at vertex C. Display dynamic text showing the side lengths and the three angle measures (α, β, γ). Construct a reference line determined by two of the triangle’s vertices, then draw several lines perpendicular to this reference line through selected auxiliary points. Compute intersection points among these perpendicular lines and the reference line to place a set of supporting points. Connect some of these supporting points to create additional lines, and continue taking intersections between the new lines and the perpendicular/reference lines to complete a structured configuration for exploring how sides and included angles relate.
The prompt above was compiled from the original drawing commands and is for reference only.