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Slope of Perpendicular Lines via Rotation

Slope of Perpendicular Lines via Rotation
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2016-10-07 20:15
VectorCoordinate GeometryTransformation
In collections A1.ACE.1 - Create and solve equations and inequalities Slope
This example explores the slope relationships of perpendicular lines by rotating a triangle. Key mathematical concepts include straight lines, rotation, vectors, and angles. Using GeoGebra commands such as Circle, Rotate, Line, Intersect, etc., the nature of perpendicular line slopes is visually demonstrated. Suitable for middle school and above students or teachers.

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