MathPlotAI Logo MathPlotAI

Unit Circle and Fraction Correspondence via Geometry

Unit Circle and Fraction Correspondence via Geometry
Loading the math board and drawing, please wait…
2021-04-02 15:47
CircleAngleTrigonometrySequenceGeometric Construction
This example demonstrates how angles on the unit circle can be geometrically linked to “fraction-like” relationships by using projections, chord/segment lengths, and constructed ratios. It centers on the unit circle, angle concepts (center vs. inscribed angles), and the way equal divisions of the circle yield points that encode fractional correspondences. Construction highlights in GeoGebra include: creating the circle and key points with Circle and Point; generating a set of division points using Sequence; drawing connecting lines and segments with Line and Segment, locating Midpoint, and building perpendiculars with OrthogonalLine; using Angle to visualize and measure the governing angular relationships; using CircleArc and CircleSector to render arc/sector structure, and Polygon/PolyLine to depict connected geometric paths; applying Min and ABS for extremum/value constraints; and using Translate, Take, and Extend for interactive transformations and extensions. The example is aimed at secondary to early undergraduate learners for visualizing unit-circle geometry and understanding how fractional relationships arise from rigorous construction.

GeoGebra Commands & Parameters

  1. Circle(A, R) → a
  2. Point(a) → E
  3. Sequence(A + (i, 0), i, R, R) → Pointdebut
  4. Line(B, yAxis) → l
  5. Point(l) → I
  6. Angle(B, A, E) → β
  7. Min(Max[betaRegularized(2, 2.3, Extend), 0.00010], 0.9999) → Animat1
  8. Zip(Circle[C, B], B, Pointdebut) → Cercles
  9. Zip(Rotate[B, ((((360 * ° / (((2 * Radius[K]) * π))) * 2) * Distance[A, B]) * π), C], K, Cercles, B, Pointdebut) → PointFins
  10. Element(PointFins, Length[PointFins]) → B'
  11. Angle(B, C, B') → α
  12. Sequence(Rotate[B, i, C], i, 0, α, α / m) → liste2
  13. Translate(liste2, Vector[B, I]) → liste2'
  14. Sequence(Polygon[Element[liste2', i], Element[liste2', i + 1], Element[liste2', i + 1] - (0, (0.275 * Extend)), Element[liste2', i] - (0, (0.275 * Extend))], i, 1, Length[liste2]) → l7
  15. Take(l7, 1, nb) → l8
  16. Segment(B, B') → b
  17. Midpoint(B, B') → D
  18. OrthogonalLine(D, b) → c
  19. Segment(A, B') → g
  20. Segment(B, B') → h
  21. Sequence(Rotate[B, i, A], i, 0, 6.283185307179586, (2 * π / m)) → liste1
  22. PolyLine(Append[B, PointFins]) → ABS
  23. CircleArc(C, B, B') → d
  24. Sequence(Segment[A, Element[liste1, i]], i, 1, Length[liste1]) → liste4
  25. Sequence(Text[k - 1 + "π/" + m, Element[liste2, k] - (0.1, 0.2)], k, 1, (2 * m) + 1) → liste3
  26. Sequence(Text["" + (k) + " π / " + m, A - (0.15, 0) + (R + 0.15; (k * π / m))], k, 2, (2 * m) - 1, 2) → liste5
  27. CircleSector(A, B, Element[liste1, 2]) → e
  28. Sequence(CircleSector[A, Element[liste1, i], Element[liste1, i + 1]], i, 1, Length[liste1]) → l3
  29. Take(l3, 1, nb) → l4
  30. Element(l4, A1) → B1
  31. Element(l4, A2) → B2
  32. Element(l4, A3) → B3
  33. Element(l4, A4) → B4
  34. Element(l4, A5) → B5
  35. Element(l4, A6) → B6
  36. Element(l4, A7) → B7
  37. Element(l4, A8) → B8
  38. Element(l4, A9) → B9
  39. Element(l4, A10) → B10
  40. Element(l4, A11) → B11
  41. Element(l4, A12) → B12
  42. Element(l4, A13) → B13
  43. Element(l4, A14) → B14
  44. Element(l4, A15) → B15
  45. Element(l4, A16) → B16
  46. Element(l4, A17) → B17
  47. Sequence(Segment[Element[liste2, i], Element[liste2, i + 1]], i, 1, Length[liste2]) → l5
  48. Take(l5, 1, nb) → l6
  49. Element(l6, A1) → C1
  50. Element(l6, A2) → C2
  51. Element(l6, A3) → C3_{1}
  52. Element(l6, A4) → C4_{1}
  53. Element(l6, A5) → C5_{1}
  54. Element(l6, A6) → C6_{1}
  55. Element(l6, A7) → C7_{1}
  56. Element(l6, A8) → C8_{1}
  57. Element(l6, A9) → C9_{1}
  58. Element(l6, A10) → C10_{1}
  59. Element(l6, A11) → C11_{1}
  60. Element(l6, A12) → C12_{1}
  61. Element(l6, A13) → C13_{1}
  62. Element(l6, A14) → C14_{1}
  63. Element(l6, A15) → C15_{1}
  64. Element(l6, A16) → C16_{1}
  65. Element(l6, A17) → C17_{1}
  66. Element(l6, A3) → C3_{2}
  67. Element(l6, A4) → C4_{2}
  68. Element(l6, A5) → C5_{2}
  69. Element(l6, A6) → C6_{2}
  70. Element(l6, A7) → C7_{2}
  71. Element(l6, A8) → C8_{2}
  72. Element(l6, A9) → C9_{2}
  73. Element(l6, A10) → C10_{2}
  74. Element(l6, A11) → C11_{2}
  75. Element(l6, A12) → C12_{2}
  76. Element(l6, A13) → C13_{2}
  77. Element(l6, A14) → C14_{2}
  78. Element(l6, A15) → C15_{2}
  79. Element(l6, A16) → C16_{2}
  80. Element(l6, A17) → C17_{2}
  81. Element(l6, A3) → C3_{3}
  82. Element(l6, A4) → C4
  83. Element(l6, A5) → C5
  84. Element(l6, A6) → C6
  85. Element(l6, A7) → C7
  86. Element(l6, A8) → C8
  87. Element(l6, A9) → C9
  88. Element(l6, A10) → C10
  89. Element(l6, A11) → C11
  90. Element(l6, A12) → C12
  91. Element(l6, A13) → C13
  92. Element(l6, A14) → C14
  93. Element(l6, A15) → C15
  94. Element(l6, A16) → C16
  95. Element(l6, A17) → C17
  96. Element(l6, A3) → C3
  97. Sequence(Text[If[i ≟ 0, "0", If[i ≟ m, " ", "\frac{" + (i / GCD[i, m]) + ("}{" + (m / GCD[i, m]) + "}")]], A - (0.1, -0.125) + (R + 0.5; ((i * 2) * π / m)), true, true], i, 1, m - 1) → l1b
  98. Sequence(Text["\frac{" + (k) + "}{" + m + "}", A - (0.085, -0.12) + (R + 0.165; ((k * 2) * π / m)), true, true], k, 1, m - 1) → l2
  99. Sequence(Text["\frac{" + (k) + "}{" + m + "}", Element[liste2, k + 1] - (0.095, -0.375), true, true], k, 1, (2 * m)) → l2b
  100. Sequence(Text[If[i ≟ 0, "0", If[i ≟ m, "", "\frac{" + (i / GCD[i, m]) + ("}{" + (m / GCD[i, m]) + "}")]], liste2(i + 1) - (0.075, 0.1), true, true], i, 1, (2 * m)) → l1
  101. CircleSector(A, Element[liste1, Length[liste1] - 1], B) → k
  102. Segment(Element[liste2, 2], B) → i
  103. Segment(Element[liste2, Length[liste2] - 1], Element[liste2, Length[liste2]]) → j
  104. Element(l4, A18) → B18
  105. Element(l4, A19) → B19
  106. Element(l4, A20) → B20
  107. Element(l4, A21) → B21
  108. Element(l4, A22) → B22
  109. Element(l4, A23) → B23
  110. Element(l4, A24) → B24
  111. Element(l6, A18) → C18
  112. Element(l6, A19) → C19
  113. Element(l6, A20) → C20
  114. Element(l6, A21) → C21
  115. Element(l6, A22) → C22
  116. Element(l6, A23) → C23
  117. Element(l8, A1) → D1
  118. Element(l6, A24) → C24
  119. Element(l8, A2) → D2
  120. Element(l8, A3) → D3
  121. Element(l8, A4) → D4
  122. Element(l8, A5) → D5
  123. Element(l8, A6) → D6
  124. Element(l8, A7) → D7
  125. Element(l8, A8) → D8
  126. Element(l8, A9) → D9
  127. Element(l8, A10) → D10
  128. Element(l8, A11) → D11
  129. Element(l8, A12) → D12
  130. Element(l8, A13) → D13
  131. Element(l8, A14) → D14
  132. Element(l8, A15) → D15
  133. Element(l8, A16) → D16
  134. Element(l8, A17) → D17
  135. Element(l8, A18) → D18
  136. Element(l8, A19) → D19
  137. Element(l8, A20) → D20
  138. Element(l8, A21) → D21
  139. Element(l8, A22) → D22
  140. Element(l8, A23) → D23
  141. Polygon(Element[liste2', 1], Element[liste2', 1 + 1], Element[liste2', 1 + 1] - (0, (0.275 * Extend)), Element[liste2', 1] - (0, (0.275 * Extend))) → q1, n, p, q, r
  142. Polygon(Element[liste2', Length[liste2'] - 1], Element[liste2', Length[liste2']], Element[liste2', Length[liste2']] - (0, (0.275 * Extend)), Element[liste2', Length[liste2'] - 1] - (0, (0.275 * Extend))) → q2, s, t, f_1, g_1

AI Drawing Prompt Reference

Draw a circle centered at a given point with a chosen radius R, using one on-circle point as a key reference. Construct another reference line and a marked angle formed by the center, the reference point, and the on-circle point. Generate an ordered set of division points on the circle by stepping through equal rotational increments, then connect consecutive points with segments to form a broken polyline/chain that visualizes fractional correspondence. From selected endpoints, draw segments and chords, and render corresponding arc and sector shapes based on the same angle and center. Add visual angle labeling for the main relationships and include the main arc segments implied by the constructed endpoints.
The prompt above was compiled from the original drawing commands and is for reference only.