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Rotational Symmetry in Action: Kite with Incircle

Rotational Symmetry in Action: Kite with Incircle
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2017-12-25 11:03
CirclePolygonTrigonometryTransformationGeometric Construction
① This example dynamically demonstrates rotational symmetry using a kite shape combined with an incircle, allowing users to observe how the figure coincides with itself after rotation about its center. ② Core mathematical concepts include rotational symmetry, symmetry axes, incircles, right triangles, trigonometry (angle calculations), and the use of equations and sequences. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as SelectedIndex, Line, Circle, Point, Intersect, Angle, If, Sequence, DynamicCoordinates, Join, Polygon, and Segment are used to build multiple points (e.g., A, B, C, P), segments, polygons (PolyPtA etc.), and dynamic coordinates, with angles α, β and sliders controlling rotation for an interactive symmetry display. ④ Suitable for middle or high school geometry classes or enrichment activities, this example helps students explore properties of rotational symmetry, relationships between incircles and polygons, and dynamic geometric construction using GeoGebra.

GeoGebra Commands & Parameters

  1. SelectedIndex(Bahasa) → nn
  2. Line(O, yAxis) → f
  3. Circle(O, Rd) → c
  4. Point(c) → C
  5. Intersect(c, f, 2) → A
  6. Angle(C, O, A) → α
  7. If(round(α / ° / (360 / a)) ≟ 0, a, round(α / ° / (360 / a))) → Near
  8. Sequence(Rotate[A, (((-n) * 360 / a))°, O], n, 1, a) → Pt0
  9. If(Distance[C, Pt0(Near)] < 0.15, Pt0(Near), C) → P
  10. DynamicCoordinates(C, x(P), y(P)) → B
  11. Circle(Rotate[B, (360 / a)°, O], Jarak) → e
  12. Circle(B, Jarak) → d
  13. Intersect(d, e) → F, G, ,
  14. If(Distance[O, F] < Distance[O, G], F, G) → H
  15. If(nn ≟ 1, {"Shape 1", "Shape 2", "Shape 3", "Shape 4", "Shape 5", "Shape 6"}, {"Bentuk 1", "Bentuk 2", "Bentuk 3", "Bentuk 4", "Bentuk 5", "Bentuk 6"}) → Bentuk
  16. SelectedIndex(Bentuk) → m
  17. Sequence(Rotate[H, ((n * 360 / a))°, O], n, 1, a) → Pt2
  18. Sequence(Rotate[B, ((n * 360 / a))°, O], n, 1, a) → Pt1
  19. Join(Sequence[{Pt1(n), Pt2(n)}, n, 1, a]) → PolyPtA
  20. Polygon(PolyPtA) → Bentuk1
  21. Segment(O, A) → h
  22. Angle(B, O, A) → β
  23. Segment(O, B) → j
  24. Join(Sequence[{Pt1(n), Pt1(n + 1), Pt2(n), Pt2(a - n + 1)}, n, 1, a, 2]) → PolyPtC
  25. Polygon(PolyPtC) → Bentuk3
  26. Join(Sequence[{Pt1(n), Pt1(n + 1), Pt2(n + 1)}, n, 1, a, 2]) → PolyPtB
  27. Polygon(PolyPtB) → Bentuk2
  28. Join(Sequence[{Pt1(n), Pt2(a - n + 1)}, n, 1, a]) → PolyPtE
  29. Polygon(PolyPtE) → Bentuk5
  30. Join(Sequence[{Pt1(n), Pt1(n + 1), Pt2(n)}, n, 1, a, 2]) → PolyPtD
  31. Polygon(PolyPtD) → Bentuk4
  32. If(Distance[O, F] < Distance[O, G], G, F) → J
  33. Sequence(Rotate[J, ((n * 360 / a))°, O], n, 1, a) → Pt3
  34. Join(Sequence[{Pt1(n), Pt2(n), Pt3(n), Pt2(n + 1)}, n, 1, a, 2]) → PolyPtF
  35. Polygon(PolyPtF) → Bentuk6

AI Drawing Prompt Reference

Draw a dynamic geometric figure demonstrating rotational symmetry: start with a circle centered at a point, place a point on the circle, and connect the center to two circle points to form an angle. Rotate this point around the center by equal steps to generate multiple symmetric points, then connect these points to form several overlapping polygons (like kites and triangles). The figure shows multiple layers of rotated shapes that coincide after rotation, with angle labels, illustrating rotational symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.