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Angle Formed by Chord & Tangent (II)

Angle Formed by Chord & Tangent (II)
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2016-04-26 09:19
CircleAngleVectorGeometric Construction
In collections Circles (Theorems)
This GeoGebra applet interactively demonstrates the measurement of an angle formed by a chord and a tangent within a circle, providing an informal proof. Key mathematical concepts include circle geometry, properties of chords and tangents, angle measurement, and geometric proofs. The construction utilizes commands such as Circle, Point, Segment, OrthogonalLine, Rotate, CircleArc, Intersect, CircleSector, Dilate, Line, Ray, Vector, Translate, and Angle, with various points and lines defined to create the dynamic figure. It is suitable for high school geometry classes to enhance student understanding.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Segment(A, C) → a
  4. OrthogonalLine(C, a) → b
  5. Rotate(C, °, A) → C'
  6. CircleArc(A, C', C) → d
  7. Point(d) → D
  8. Segment(D, C) → e
  9. Circle(C, (0.2 * a)) → g
  10. Intersect(g, b, 2) → E
  11. Intersect(g, e, 1) → F
  12. Segment(A, C) → h
  13. Segment(A, D) → i
  14. CircleSector(C, E, F) → k
  15. Dilate(A, If[0 < j ≤ 1, 1 - j, 0], C) → A'
  16. Dilate(A, If[0 < j ≤ 1, 1 - j, 0], D) → A'_1
  17. Line(A, e) → n
  18. Dilate(C, If[3 < j ≤ 4, 4 - j, 0], A) → C'_2
  19. CircleArc(A, C, D) → q
  20. Point(b) → J
  21. Ray(C, J) → r
  22. CircleArc(A, D, C) → p
  23. Point(p) → K
  24. Circle(K, (0.2 * a)) → s
  25. Segment(K, C) → t
  26. Segment(K, D) → a_1
  27. Intersect(s, t, 1) → L
  28. Intersect(s, a_1, 1) → M
  29. Dilate(K, If[0 < j ≤ 1, 1 - j, 0], C) → K'
  30. Dilate(K, If[0 < j ≤ 1, 1 - j, 0], D) → K'_1
  31. Segment(K, K') → f
  32. Segment(K, K'_1) → l
  33. Dilate(C, If[1 < j ≤ 2, 2 - j, 0], K) → C'_1
  34. Vector(C, C'_1) → u
  35. Translate(k, u) → k'
  36. Translate(E, u) → E'
  37. Angle(L, K, E') → α
  38. Rotate(k', If[2 < j ≤ 3, ((j - 2) * ((2 * π) - α)), (2 * π) - α], K) → k''