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Effects of Dilations on Angles

Effects of Dilations on Angles
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2019-06-26 09:10
AngleTransformationGeometric Construction
This interactive applet helps students discover why angles remain unchanged under similarity transformations. Key concepts include similarity transformations and angle invariance. It uses commands like polygon, vector, and angle in GeoGebra. Suitable for middle school and above students to learn the basic properties of similarity transformations.

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

AI Drawing Prompt Reference

Draw a triangle and a second triangle obtained by dilating the first about an external center point, with a variable scale factor controlled by an input box or slider. Connect the center to corresponding vertices with dashed lines to show the dilation. Place small circles at corresponding vertices and mark the angles with circular sectors to illustrate that angles remain unchanged under the transformation. Include dynamic arrows from original vertices to their dilated images to demonstrate the invariance of angles in a similarity transformation.
The prompt above was compiled from the original drawing commands and is for reference only.