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Optimization of Sum of Areas of Two Triangles Between Parallel Lines

Optimization of Sum of Areas of Two Triangles Between Parallel Lines
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2016-10-22 09:18
TriangleCoordinate GeometryCalculusOptimizationGeometric Construction
This example demonstrates how to position two triangles between two parallel lines to minimize the sum of their areas. Key math concepts include area calculation, calculus optimization, and trigonometric function applications. GeoGebra commands and objects used in construction are Rotate, Angle, Line, Circle, Intersect, Ray, Point, OrthogonalLine, Segment, Polygon, CircleSector, Dilate, Vector, and Translate. Suitable for high school and above students learning calculus optimization and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Rotate(B_1, α, A) → B'
  2. Angle(B_1, A, B') → β
  3. Line(A, B') → f
  4. Circle(A, 1) → c
  5. Intersect(c, f, 2) → C_1
  6. Intersect(c, f, 1) → D_1
  7. Ray(A, C_1) → g
  8. Ray(A, D_1) → h
  9. Point(g) → B
  10. Circle(A, a) → d
  11. OrthogonalLine(A, g) → i
  12. Intersect(d, i, 2) → F
  13. Intersect(d, i, 1) → G
  14. Line(A, F) → j
  15. Line(F, g) → k
  16. Rotate(B, γ, A) → E'
  17. Angle(B, A, E') → δ
  18. Ray(A, E') → l
  19. Intersect(k, l) → C
  20. Circle(C, 1) → e
  21. Intersect(e, k, 1) → I
  22. Intersect(e, k, 2) → J
  23. Ray(C, I) → m
  24. Ray(C, J) → n
  25. Point(m) → D
  26. Segment(B, D) → p
  27. Intersect(l, p) → L
  28. Segment(A, C) → q
  29. Polygon(A, B, L) → poly1, l_1, a_1, e_1
  30. Polygon(L, C, D) → poly2, k_1, l_2, h_1
  31. Intersect(g, xAxis) → M
  32. Intersect(g, xAxis) → N
  33. Segment(D, C) → s
  34. Segment(A, B) → t
  35. Segment(A, L) → f_1
  36. Segment(L, C) → g_1
  37. Circle(A, Min[(0.35 * f_1), (0.35 * g_1)]) → c_1
  38. Intersect(c_1, t) → O,
  39. Intersect(c_1, f_1) → P,
  40. CircleSector(A, O, P) → d_1
  41. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], C) → A'
  42. Vector(A, A') → v
  43. Translate(d_1, v) → p_1
  44. Rotate(p_1, If[1 < u ≤ 2, ((u - 1) * π), π], C) → q_1