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Rotation and Similarity in Equilateral Triangles

Rotation and Similarity in Equilateral Triangles
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2017-11-18 09:13
TriangleGeometric Construction
This example illustrates rotations and similarity transformations in an equilateral triangle. Key mathematical concepts include rotation, similar triangles, and geometric construction. Using GeoGebra commands such as Rotate, Dilate, and Polygon, dynamic demonstrations of the changes in an equilateral triangle under rotation and similarity transformations are provided. Suitable for middle school and above students to learn about geometric transformations and similar triangles.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(C, B) → i
  8. Polygon(A, B, C) → t1, c, a_1, b
  9. Circle(A, (0.15 * Min[h, f])) → d
  10. Intersect(d, f) → D,
  11. Intersect(d, h) → E,
  12. CircleSector(A, D, E) → e
  13. Rotate(B, π / 3, C) → B'_1
  14. Rotate(C, π / 3, A) → C'
  15. Rotate(B, π / 3, A) → B'_2
  16. Rotate(A, If[0 < j ≤ 1, (j * π / 3), π / 3], A) → A'
  17. Rotate(C, If[0 < j ≤ 1, (j * π / 3), π / 3], A) → C'_1
  18. Segment(A', C'_1) → h'
  19. Rotate(C, If[0 < j ≤ 1, (j * π / 3), π / 3], C) → C'_2
  20. Rotate(B, If[0 < j ≤ 1, (j * π / 3), π / 3], C) → B'_3
  21. Segment(C'_2, B'_3) → i'
  22. Rotate(A, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], C) → A'_1
  23. Rotate(C, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], C) → C'_3
  24. Segment(A'_1, C'_3) → h'_1
  25. Rotate(C, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], B) → C'_4
  26. Rotate(B, If[1 < j ≤ 2, ((-(j - 1)) * π / 3), ((-π)) / 3], B) → B'_4
  27. Segment(C'_4, B'_4) → i'_1
  28. Polygon(A, C, C'_1) → t2, c'_1, a_2, c_1
  29. Polygon(B, B'_3, C) → t3, c_2, b_1, b'_3
  30. Rotate(A, If[2 < j ≤ 3, ((j - 2) * π / 3), π / 3], A) → A'_2
  31. Rotate(B, If[2 < j ≤ 3, ((j - 2) * π / 3), π / 3], A) → B'_5
  32. Segment(A'_2, B'_5) → f'
  33. Rotate(A, If[3 < j ≤ 4, ((-(j - 3)) * π / 3), ((-π)) / 3], B) → A'_3
  34. Rotate(B, If[3 < j ≤ 4, ((-(j - 3)) * π / 3), ((-π)) / 3], B) → B'_6
  35. Segment(A'_3, B'_6) → f'_1
  36. Dilate(C'_1, If[5 < j ≤ 6, 6 - j, 0], B'_5) → F
  37. Vector(C'_1, F) → u
  38. Translate(A'_1, u) → G
  39. Translate(C'_3, u) → H
  40. Segment(G, H) → k
  41. Dilate(C, If[6 < j ≤ 7, 7 - j, 0], C'_1) → C'_5
  42. Vector(C, C'_5) → v
  43. Translate(C'_2, v) → I
  44. Translate(B'_3, v) → J
  45. Segment(I, J) → i''
  46. Polygon(C'_1, C, B'_3, B'_5) → q1, c'_2, c_3, b'_1, b'_5