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Exploring Slopes of Perpendicular Lines via Rotation

Exploring Slopes of Perpendicular Lines via Rotation
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2016-07-14 09:03
AngleCoordinate GeometryTransformation
In collections A1.ACE.1 - Create and solve equations and inequalities Expressing Geometric Properties with Equations (GPE) Cluster Parallel Lines and Planes Slope
This example explores the relationship between the slopes of perpendicular lines through rotation. Key concepts include slope, rotation, and angles. Commands used include Circle, Rotate, Line, Intersect, OrthogonalLine, Segment, Dilate, etc. Suitable for middle school and above students for geometric exploration and function graphing.

GeoGebra Commands & Parameters

  1. Circle(A, a_2) → c
  2. Rotate(B, α, A) → B'
  3. Line(A, B') → j
  4. Rotate(j, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → k
  5. Intersect(c, j, 2) → C
  6. OrthogonalLine(C, g) → h
  7. Intersect(g, h) → D
  8. Line(D, j) → i_1
  9. Intersect(k, i_1) → P
  10. Segment(C, D) → l
  11. Segment(A, D) → j_1
  12. Dilate(C, If[0 < b_1 ≤ 1, 1 - b_1, 0], D) → C'
  13. Segment(C, C') → b
  14. Min(b, j_1) → k_2
  15. Circle(D, (0.2 * k_2)) → e
  16. Intersect(e, l, 1) → F
  17. Angle(F, D, P) → δ
  18. Intersect(e, g, 2) → E
  19. CircleSector(D, E, F) → p
  20. Rotate(p, If[2 < b_1 ≤ 3, ((b_1 - 2) * π / 2), π / 2], D) → p'
  21. Angle(B, A, B') → β
  22. Polygon(A, D, C) → poly1, c_1, a_1, d
  23. Point(g) → G
  24. Ray(D, G) → m
  25. Point(m) → H
  26. Ray(D, H) → n
  27. Dilate(D, If[1 < b_1 ≤ 2, 2 - b_1, 0], A) → D'
  28. Segment(D, D') → a
  29. Ray(D, I) → i
  30. Intersect(e, i, 1) → J
  31. CircleSector(D, F, J) → q
  32. Circle(C, (0.2 * k_2)) → r
  33. Circle(A, (0.2 * k_2)) → s
  34. Intersect(s, j_1, 1) → K
  35. Intersect(s, j, 2) → L
  36. Intersect(r, l, 1) → M
  37. Intersect(r, j, 1) → N
  38. CircleSector(C, N, M) → t
  39. CircleSector(A, K, L) → d_1
  40. Rotate(q, If[2 < b_1 ≤ 3, ((-(b_1 - 2)) * π / 2), ((-π)) / 2], D) → q'
  41. CircleSector(C, M, N) → e_1
  42. CircleSector(A, L, K) → f_1
  43. Rotate(A, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → A'
  44. Rotate(D, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → D'_1
  45. Rotate(C, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → C'_1
  46. Polygon(A', D'_1, C'_1) → poly1', c'_1, a', d'_1
  47. Rotate(C, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → C'_2
  48. Rotate(C', If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → C''
  49. Segment(C'_2, C'') → b'
  50. Rotate(D, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → D'_2
  51. Rotate(D', If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → D''
  52. Segment(D'_2, D'') → a'_1
  53. Rotate(d_1, If[4.5 < b_1 ≤ 5.5, ((b_1 - 4.5) * π / 2), π / 2], A) → g_1
  54. CircleArc(A, K, L) → h_1
  55. Rotate(p', If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → p''
  56. Rotate(t, If[5.5 < b_1 ≤ 6.5, ((-(b_1 - 5.5)) * π), (-π)], C) → t'
  57. CircleArc(C, N, M) → k_1
  58. Dilate(C, If[6.5 < b_1 ≤ 7.5, 7.5 - b_1, 0], A) → C'_3
  59. Vector(C, C'_3) → u
  60. Translate(t', u) → t''
  61. Dilate(D, If[9 < b_1 ≤ 10, 10 - b_1, 0], A) → D'_3
  62. Vector(D, D'_3) → v
  63. Rotate(p', If[8 < b_1 ≤ 9, ((b_1 - 8) * δ), δ], D) → p''_1
  64. Translate(p''_1, v) → p_1
  65. Point(k) → O
  66. Rotate(C, If[3.5 < b_1 ≤ 4.5, ((b_1 - 3.5) * π / 2), π / 2], A) → C'_4