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Animation 117: Geometric Transformations with Circles and Lines

Animation 117: Geometric Transformations with Circles and Lines
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2016-09-19 05:13
CircleAngleTransformationGeometric ConstructionSlider Animation
①This GeoGebra example presents a dynamic geometric animation composed of circles, lines, rays, and sectors, demonstrating how rotations and dilations affect the interrelations among various parts of the figure. ②Core mathematical concepts include angles, circles, straight lines, planes, isosceles triangles, fractions, and basic statistics, along with the interplay between algebraic and geometric objects. ③Construction highlights: from the summary, commands such as Intersect, Ray, Rotate, Angle, Circle, CircleSector, Segment, Point, CircleArc, and Dilate are used. Objects include multiple angles (α, β, γ), numerous points (A, B, C, D, E, F, G, H, I, J, K), lines (f, g, h), circle c, segments (a, b, d, e), and derived points like B', F', H' created via rotation and dilation. ④This is suitable for middle and high school geometry classes to explore rotation symmetry, properties of isosceles triangles, circle-line intersections, and the effects of parameter changes in dynamic constructions, also useful for inquiry-based learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Ray(A, B) → f
  3. Rotate(B, α, A) → B'
  4. Angle(B, A, B') → β
  5. Ray(A, B') → g
  6. Circle(A, 1) → c
  7. Rotate(B, α / 2, A) → B'_1
  8. Angle(B, A, B'_1) → γ
  9. Ray(A, B'_1) → h
  10. Intersect(c, h, 1) → C
  11. Intersect(c, f, 1) → D
  12. CircleSector(A, D, C) → d
  13. Intersect(c, g, 1) → E
  14. CircleSector(A, D, E) → e
  15. Circle(A, a) → k
  16. Circle(A, b) → p
  17. Intersect(k, f, 1) → F
  18. Intersect(k, g, 1) → G
  19. Intersect(p, f, 1) → H
  20. Intersect(p, g, 1) → I
  21. Segment(F, I) → i
  22. Segment(H, G) → j
  23. Intersect(h, i) → J
  24. Point(h) → K
  25. Ray(J, K) → l
  26. CircleSector(A, H, I) → q
  27. Rotate(F, If[0 < m ≤ 1, (m * α), α], A) → F'
  28. CircleArc(A, F, F') → r
  29. Rotate(H, If[1 < m ≤ 2, ((m - 1) * α), α], A) → H'
  30. CircleArc(A, H, H') → s
  31. Dilate(F, If[2 < m ≤ 3, 3 - m, 0], I) → F'_1
  32. Segment(F, F'_1) → n
  33. Dilate(H, If[3 < m ≤ 4, 4 - m, 0], G) → H'_1
  34. Segment(H, H'_1) → t
  35. Dilate(A, If[5 < m ≤ 6, 6 - m, 0], J) → A'
  36. Segment(A, A') → f_1
  37. Rotate(d, If[7 < m ≤ 8, ((m - 7) * α / 2), α / 2], A) → d'