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Exterior Angles of Polygons Demo by Anthony OR 柯志明

Exterior Angles of Polygons Demo by Anthony OR 柯志明
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2011-09-15 17:03
PolygonAngleVectorGeometric ConstructionSlider Animation
In collections GCSE Higher Ch5 Angles & Trigonometry Angles in polygons PDGE Year 1: Making Use of ICT Resources
This example visually demonstrates the properties of exterior angles in polygons by simulating a car turning at each vertex. It covers core concepts like exterior angles, interior angles, and the sum of interior angles theorem. GeoGebra commands used include Rotate, Polygon, Vector, and Angle. Suitable for middle school and above students to grasp spatial geometry relationships.

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. Midpoint(a_1) → O
  22. Segment(A, A + ((1.5 * r) * UnitVector[Segment[E, A]])) → g
  23. Segment(B, B + ((1.5 * r) * UnitVector[Segment[A, B]])) → h
  24. Segment(C, C + ((1.5 * r) * UnitVector[Segment[B, C]])) → i
  25. Segment(D, D + ((1.5 * r) * UnitVector[Segment[C, D]])) → j
  26. Segment(E, E + ((1.5 * r) * UnitVector[Segment[D, E]])) → k
  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. CircleSector(C, If[c ≤ 180°, C + (r * UnitVector[Segment[B, C]]), C + (r * UnitVector[Segment[C, D]])], If[c ≤ 180°, D, C + (r * UnitVector[Segment[B, C]])]) → c'
  33. CircleSector(A, If[a ≤ 180°, A + (r * UnitVector[Segment[E, A]]), A + (r * UnitVector[Segment[A, B]])], If[a ≤ 180°, B, A + (r * UnitVector[Segment[E, A]])]) → a'
  34. CircleSector(B, If[b ≤ 180°, B + (r * UnitVector[Segment[A, B]]), B + (r * UnitVector[Segment[B, C]])], If[b ≤ 180°, C, B + (r * UnitVector[Segment[A, B]])]) → b'
  35. CircleSector(D, If[d ≤ 180°, D + (r * UnitVector[Segment[C, D]]), D + (r * UnitVector[Segment[D, E]])], If[d ≤ 180°, E, D + (r * UnitVector[Segment[C, D]])]) → d'
  36. CircleSector(E, If[e ≤ 180°, E + (r * UnitVector[Segment[D, E]]), E + (r * UnitVector[Segment[E, A]])], If[e ≤ 180°, A, E + (r * UnitVector[Segment[D, E]])]) → e'
  37. Corner(1) → I
  38. Corner(3) → J
  39. Corner(5) → pixelnos
  40. If(s ≤ 0.5, (-8 * (1 - (2 * s))) - (80 * s), (-40 * (1 - (2 * (s - 0.5)))) - (1120 * (s - 0.5))) → xmin
  41. If(s ≤ 0.5, (-5 * (1 - (2 * s))) - (50 * s), (-25 * (1 - (2 * (s - 0.5)))) - (700 * (s - 0.5))) → ymin
  42. If(a + b + c + d + e ≟ 360°, true, false) → convex