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Rotating from Parallel to Perpendicular

Rotating from Parallel to Perpendicular
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2020-01-14 22:32
AngleTransformationGeometric ConstructionSlider AnimationLocus
In collections The variable-parts perspective across the curriculum
① This example dynamically rotates a line to show its transition from being parallel to another line to becoming perpendicular, with real-time updates of angles and intersection points. ② Core mathematical concepts include parallel and perpendicular line relationships, angle measurement during rotation, and the coordinated changes of geometric objects (intersections, perpendiculars, circles) as rotation proceeds. ③ Construction highlights: Using GeoGebra commands such as Point, Intersect, Rotate, Angle, Ray, OrthogonalLine, Circle, Line, Segment, and Polygon, the construction includes line a, rotation angle α, perpendicular line f, circle c, and multiple auxiliary points (A, B, C, D, etc.) and segments, with rotation controlled by a slider or angle parameter to update the figure live. ④ Suitable for middle and high school geometry classes, this helps students understand rotation transformations, criteria for parallel and perpendicular lines, and the visualization of angle-position relationships in dynamic geometry, and is also useful for teachers creating interactive lessons.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Rotate(A, α, B) → A'
  4. Angle(A, B, A') → β
  5. Ray(B, A') → f
  6. OrthogonalLine(B, f) → g
  7. Circle(B, 0.7) → c
  8. Intersect(c, g, 2) → C
  9. Intersect(c, yAxis, 1) → D
  10. Circle(D, 0.5) → d
  11. Circle(C, 0.5) → e
  12. Intersect(e, g, 2) → E
  13. Intersect(d, yAxis, 1) → F
  14. Line(D, xAxis) → h
  15. Line(F, h) → i
  16. Line(C, f) → j
  17. Line(E, f) → k
  18. Circle(F, a) → p
  19. Circle(E, a) → q
  20. Intersect(p, i, 2) → G
  21. Intersect(q, k, 2) → H
  22. Circle(G, a) → r
  23. Intersect(r, i, 2) → I
  24. Circle(I, a) → s
  25. Intersect(s, i, 2) → J
  26. Circle(J, a) → t
  27. Intersect(t, i, 2) → K
  28. Circle(H, a) → c_1
  29. Intersect(c_1, k, 2) → L
  30. Circle(L, a) → d_1
  31. Intersect(d_1, k, 2) → M
  32. Circle(F, 0.3) → e_1
  33. Circle(E, 0.3) → f_1
  34. Intersect(e_1, yAxis, 1) → N
  35. Intersect(f_1, g, 2) → O
  36. Line(O, k) → l
  37. Line(N, i) → m
  38. Line(M, g) → n
  39. OrthogonalLine(K, i) → b
  40. Intersect(m, b) → P
  41. Intersect(l, n) → Q
  42. Intersect(j, n) → R
  43. Intersect(h, b) → S
  44. Line(L, n) → g_1
  45. Line(H, g_1) → h_1
  46. Line(J, b) → i_1
  47. Line(I, b) → j_1
  48. Line(G, b) → k_1
  49. Intersect(j, h_1) → T
  50. Intersect(j, g_1) → U
  51. Intersect(h, k_1) → V
  52. Intersect(h, j_1) → W
  53. Intersect(h, i_1) → Z
  54. Segment(N, P) → l_1
  55. Segment(O, Q) → m_1
  56. Polygon(F, G, V, D) → q1, f_2, g_2, v, d_2
  57. Polygon(G, I, W, V) → q2, g_3, i_2, w, v_1
  58. Polygon(I, J, Z, W) → q3, i_3, j_2, z_1, w_1
  59. Polygon(J, K, S, Z) → q4, j_3, k_2, s_1, z_2
  60. Polygon(C, T, H, E) → q5, c_2, t_1, h_2, e_2
  61. Polygon(T, U, L, H) → q6, t_2, u, l_2, h_3
  62. Polygon(U, R, M, L) → q7, u_1, r_1, m_2, l_3

AI Drawing Prompt Reference

Draw a ray starting from the origin on a coordinate plane, rotating from parallel to the horizontal axis to perpendicular. Show the rotation angle. Construct a line perpendicular to the ray at the origin. Draw a small circle centered at the origin, intersecting the perpendicular line and the vertical axis at two points. From these points, draw two more circles, generating further intersections. Extend parallel lines horizontally and vertically through these points. Symmetrically place a series of equally spaced points along both sides of the ray, connecting them with segments to form a dynamic geometric pattern illustrating the transition from parallel to perpendicular.
The prompt above was compiled from the original drawing commands and is for reference only.