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Polygon Angles and Transformations

Polygon Angles and Transformations
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2017-01-12 05:12
PolygonAngleGeometric Construction
This example demonstrates the calculation of interior angles in polygons and rotational transformations. It involves core concepts like polygons, angles, geometric transformations, and similar triangles. Construction uses commands such as Rotate, Polygon, If, Vector, Angle, OrthogonalLine, Point, Midpoint, Translate, and Segment to build polygons and illustrate transformations. Suitable for high school mathematics teaching and geometric transformation studies.

GeoGebra Commands & Parameters

  1. Rotate((x(P1) + (x(A) - x(E)) / Distance[E, A], y(P1) + (y(A) - y(E)) / Distance[E, A]), (Min[Max[t - 1, 0], 1] * (180° - Angle[B, A, E])), P1) → P1'
  2. Rotate((x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B]), (Min[Max[t - 3, 0], 1] * (180° - Angle[C, B, A])), P2) → P2'
  3. Rotate((x(P3) + (x(C) - x(B)) / Distance[B, C], y(P3) + (y(C) - y(B)) / Distance[B, C]), (Min[Max[t - 5, 0], 1] * (180° - Angle[D, C, B])), P3) → P3'
  4. Rotate((x(P4) + (x(D) - x(C)) / Distance[C, D], y(P4) + (y(D) - y(C)) / Distance[C, D]), (Min[Max[t - 7, 0], 1] * (180° - Angle[E, D, C])), P4) → P4'
  5. Rotate((x(P5) + (x(E) - x(D)) / Distance[E, D], y(P5) + (y(E) - y(D)) / Distance[E, D]), (Min[Max[t - 9, 0], 1] * (180° - Angle[A, E, D])), P5) → P5'
  6. Polygon(E, A, B, C, D) → poly1, a_1, b_1, c_1, d_1, e_1
  7. If(t ≤ 2, P1, If[t ≤ 4, P2, If[t ≤ 6, P3, If[t ≤ 8, P4, If[t ≤ 10, P5, P6]]]]) → P
  8. If(t ≤ 2, P1', If[t ≤ 4, P2', If[t ≤ 6, P3', If[t ≤ 8, P4', If[t ≤ 10, P5', P6']]]]) → P'
  9. Vector(P, P') → u
  10. Angle(P1', P1, (x(P1) + (x(A) - x(E)) / Distance[E, A], y(P1) + (y(A) - y(E)) / Distance[E, A])) → α
  11. Angle(P2', P2, (x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B])) → β
  12. Angle(P3', P3, (x(P3) + (x(C) - x(B)) / Distance[B, C], y(P3) + (y(C) - y(B)) / Distance[B, C])) → γ
  13. Angle(P4', P4, (x(P4) + (x(D) - x(C)) / Distance[C, D], y(P4) + (y(D) - y(C)) / Distance[C, D])) → δ
  14. Angle(P5', P5, (x(P5) + (x(E) - x(D)) / Distance[E, D], y(P5) + (y(E) - y(D)) / Distance[E, D])) → ε
  15. OrthogonalLine(P, u) → f
  16. Point(f) → F
  17. Midpoint(F, P') → G
  18. Vector(G, P) → v
  19. Translate(P, v) → P'_1
  20. Translate(P', v) → P''
  21. Segment(A, (x(A) + (x(A) - x(E)) / Distance[E, A], y(A) + (y(A) - y(E)) / Distance[E, A])) → g
  22. Segment(B, (x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B])) → h
  23. Segment(C, (x(C) + (x(C) - x(B)) / Distance[B, C], y(C) + (y(C) - y(B)) / Distance[B, C])) → i
  24. Segment(D, (x(D) + (x(D) - x(C)) / Distance[C, D], y(D) + (y(D) - y(C)) / Distance[C, D])) → j
  25. Segment(E, (x(E) + (x(E) - x(D)) / Distance[D, E], y(E) + (y(E) - y(D)) / Distance[D, E])) → k
  26. Midpoint(a_1) → O
  27. Angle((x(A) + (x(A) - x(E)) / Distance[E, A], y(A) + (y(A) - y(E)) / Distance[E, A]), A, B) → a
  28. Angle((x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B]), B, C) → b
  29. Angle((x(C) + (x(C) - x(B)) / Distance[B, C], y(C) + (y(C) - y(B)) / Distance[B, C]), C, D) → c
  30. Angle((x(D) + (x(D) - x(C)) / Distance[C, D], y(D) + (y(D) - y(C)) / Distance[C, D]), D, E) → d
  31. Angle((x(E) + (x(E) - x(D)) / Distance[D, E], y(E) + (y(E) - y(D)) / Distance[D, E]), E, A) → e
  32. If((t ≥ 1) ∧ (t < 2), α, If[t ≥ 2, a, 0°]) → α'
  33. If((t ≥ 3) ∧ (t < 4), β, If[t ≥ 4, b, 0°]) → β'
  34. If((t ≥ 5) ∧ (t < 6), γ, If[t ≥ 6, c, 0°]) → γ'
  35. If((t ≥ 7) ∧ (t < 8), δ, If[t ≥ 7, d, 0°]) → δ'
  36. If((t ≥ 9) ∧ (t < 10), ε, If[t ≥ 9, e, 0°]) → ε'