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Angles in Dilation: Exploring Similarity Transformations

Angles in Dilation: Exploring Similarity Transformations
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2017-03-01 05:36
PolygonAngleTransformationGeometric Construction
This interactive applet demonstrates that angles remain invariant under dilation, proving it is a similarity transformation. Key concepts include dilation, similarity, and angle measurement. Construction utilizes GeoGebra commands such as Polygon, Dilate, Angle, and Vector to visualize the transformation. Ideal for high school geometry students exploring transformational properties.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Segment(A, B) → d
  3. Segment(B, C) → e
  4. Segment(C, A) → f
  5. Textfield(k) → InputBox1
  6. Dilate(A, k, O) → A'
  7. Dilate(B, k, O) → B'
  8. Dilate(C, k, O) → C'
  9. Polygon(A', B', C') → poly1', c', a', b'
  10. Dilate(A, k, O) → A'_1
  11. Dilate(B, k, O) → B'_1
  12. Segment(A'_1, B'_1) → d'
  13. Dilate(B, k, O) → B'_2
  14. Dilate(C, k, O) → C'_1
  15. Segment(B'_2, C'_1) → e'
  16. Dilate(C, k, O) → C'_2
  17. Dilate(A, k, O) → A'_2
  18. Segment(C'_2, A'_2) → f'
  19. Segment(O, A') → g
  20. Segment(O, A) → h
  21. Segment(O, B) → i
  22. Segment(O, B') → j
  23. Segment(O, C') → l
  24. Segment(O, C) → m
  25. Circle(A, 1) → p
  26. Circle(B, 1) → s
  27. Intersect(p, f, 1) → D
  28. Intersect(p, d, 1) → E
  29. Intersect(s, d, 1) → F
  30. Intersect(s, e, 1) → G
  31. CircleSector(A, D, E) → t
  32. CircleSector(B, F, G) → c_1
  33. Dilate(A, If[0 < n ≤ 1, 1 - n, 0], A') → A'_3
  34. Dilate(B, If[1 < n ≤ 2, 2 - n, 0], B') → B'_3
  35. Vector(B, B'_3) → u
  36. Vector(A, A'_3) → v
  37. Translate(t, v) → t'
  38. Translate(c_1, u) → d_1
  39. Angle(D, A, E) → α
  40. CircleSector(A, E, D) → e_1
  41. CircleSector(B, G, F) → f_1
  42. Translate(e_1, v) → g_1
  43. Translate(f_1, u) → h_1
  44. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], A') → A'_4
  45. Vector(A, A'_4) → w
  46. Dilate(B, If[2 < o ≤ 3, 3 - o, 0], B') → B'_4
  47. Vector(B, B'_4) → q
  48. Translate(e_1, w) → r
  49. Rotate(r, If[1 < o ≤ 2, (((o - 1) * 180))°, 180°], A') → r'
  50. Translate(f_1, q) → k_1
  51. Rotate(k_1, If[3 < o ≤ 4, (((o - 3) * 180))°, 180°], B') → p_1
  52. Translate(t, w) → t'_1
  53. Rotate(t'_1, If[1 < o ≤ 2, (((o - 1) * 180))°, 180°], A') → q_1
  54. Translate(c_1, q) → r_1
  55. Rotate(r_1, If[3 < o ≤ 4, (((o - 3) * 180))°, 180°], B') → s_1