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Law of Cosines: Discovery Investigation

Law of Cosines: Discovery Investigation
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2016-08-07 10:32
TriangleCircleAngleTrigonometryGeometric Construction
① This interactive GeoGebra example demonstrates the Law of Cosines by allowing users to manipulate a triangle and observe how the squared side lengths relate to the cosine of the included angle, leading to a discovery of the formula c² = a² + b² - 2ab·cosγ. ② Core mathematical concepts include the Law of Cosines, triangle geometry, angle measurement, vectors, circumcircles, and translations. ③ Construction highlights: using GeoGebra commands such as Rotate, Angle, Ray, Segment, Point, OrthogonalLine, Intersect, Polygon, Distance, Min, Circle, CircleSector, Dilate, and Line, the applet builds a dynamic triangle with auxiliary circles, angle bisectors, and a circumcircle, enabling real-time updates of side lengths and cosine values as vertices are dragged. ④ Suitable for high school geometry or trigonometry classes as a discovery-based learning tool, and also useful for college-level review or teacher demonstrations. The construction integrates dynamic geometry with algebraic computation, fostering an inquiry-driven understanding of the theorem.

GeoGebra Commands & Parameters

  1. Rotate(A_1, α, C) → A'
  2. Angle(A_1, C, A') → β
  3. Ray(A_1, A') → f
  4. Segment(A_1, C) → g
  5. Ray(C, A') → h
  6. Ray(C, A_1) → i
  7. Point(i) → A
  8. Point(h) → B
  9. Segment(A, B) → j
  10. Segment(C, A) → k
  11. Segment(C, B) → l
  12. OrthogonalLine(C, j) → m
  13. Intersect(m, j) → D
  14. Polygon(A, D, C) → poly1, b, c, e
  15. Polygon(D, C, B) → poly2, d, e_1, b_1
  16. Angle(C, D, A) → γ
  17. Distance(A, D) → a
  18. Distance(D, B) → n
  19. Min(a, n) → o
  20. Circle(D, (0.2 * o)) → p
  21. Segment(C, D) → q
  22. Intersect(p, j) → F, G
  23. Intersect(p, q) → H,
  24. CircleSector(D, G, H) → r
  25. Dilate(C, If[0 < s ≤ 1, 1 - s, 0], D) → B'
  26. Segment(C, B') → t
  27. Rotate(r, If[1.5 < s ≤ 2.5, ((s - 1.5) * π / 2), π / 2], D) → r'
  28. Dilate(D, If[2.5 < s ≤ 3.5, 3.5 - s, 0], A) → E'
  29. Dilate(C, If[2.5 < s ≤ 3.5, 3.5 - s, 0], A) → B'_1
  30. Segment(C, B'_1) → f_1
  31. Segment(D, E') → g_1
  32. Circle(D, 1) → c_1
  33. Intersect(c_1, m, 1) → I
  34. Line(A, m) → h_1
  35. Line(B, m) → i_1
  36. OrthogonalLine(I, m) → j_1
  37. Intersect(h_1, j_1) → J
  38. Intersect(i_1, j_1) → K
  39. Polygon(A, D, C) → poly3, c_2, a_1, d_1
  40. Dilate(D, If[2.5 < s ≤ 3.5, 3.5 - s, 0], B) → D'
  41. Dilate(C, If[2.5 < s ≤ 3.5, 3.5 - s, 0], B) → C'
  42. Segment(C, C') → k_1
  43. Segment(D, D') → l_1
  44. Dilate(A, If[3.5 < s ≤ 4.5, 4.5 - s, 0], J) → A'_1
  45. Dilate(B, If[3.5 < s ≤ 4.5, 4.5 - s, 0], K) → B'_2
  46. Segment(A, A'_1) → m_1
  47. Segment(B, B'_2) → n_1
  48. Dilate(J, If[4.5 < s ≤ 5.5, 5.5 - s, 0], K) → J'
  49. Segment(J, J') → p_1

AI Drawing Prompt Reference

Draw a dynamic triangle with one side rotated to form an included angle, illustrating the Law of Cosines. Construct a triangle, drop a perpendicular from a vertex to the opposite side, and mark the foot. Use the foot as center to draw a small circle and a sector representing the angle. Add auxiliary lines to divide the triangle into smaller parts, showing relationships between side lengths and cosine values. Use a slider to progressively reveal different stages of the construction.
The prompt above was compiled from the original drawing commands and is for reference only.