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Probability of Existence of a Rectangle

Probability of Existence of a Rectangle
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2015-02-18 02:03
PolygonProbability & StatisticsGeometric Construction
In collections Pi Approximations
This example demonstrates the probability of the existence of a rectangle given a perimeter of 2p and an area of q as integers. Key mathematical concepts include probability and geometry. Using GeoGebra commands such as Vector, Intersect, and Polygon, the conditions for the existence of a rectangle are constructed. Suitable for middle school and above students learning geometric probability.

GeoGebra Commands & Parameters

  1. Vector(((-λ), (-λ)), J) → u
  2. Intersect(f, xAxis, 1) → A
  3. Polygon(A, B, 4) → Polígon1, a_1, b, c, d, C, D
  4. Intersect(f, c, 1) → E
  5. Sequence(Sequence[(p, k), k, 1, p^(2) / 4 - 1 / 8 + (-1)^(p) / 8], p, 1, t) → llista1
  6. Sequence(Sequence[(k, h), h, 1, a], k, floor(t) + 1, a) → llista2
  7. GCD(S1, a^(2)) → j
  8. Translate(Polygon[((-λ), (-λ)), (((n * a) * λ), (-λ)), 4], u) → Polígon2
  9. Translate(fs, u) → tfs
  10. Translate(Segment[(0, (-λ)), (0, ((n * a) * λ))], u) → e2
  11. Translate(fp, u) → tfp
  12. Translate(((λ * (p + (p^(2) - (4 * q))^(0.5)) / 2), (λ * (p - (p^(2) - (4 * q))^(0.5)) / 2)), u) → F
  13. Translate(Segment[((-λ), 0), (((n * a) * λ), 0)], u) → e1
  14. OrthogonalLine(F, e1) → h
  15. OrthogonalLine(F, e2) → i
  16. Intersect(h, e1) → G
  17. Intersect(i, e2) → H
  18. Intersect(e1, e2) → I
  19. Polygon(H, F, G, I) → Polígon3, h_1, f_1, g_1, i_1
  20. Sequence(Translate[Segment[((k * λ), (-λ)), ((k * λ), ((n * a) * λ))], u], k, 1, (n * a)) → se2
  21. Sequence(Translate[Segment[((-λ), (k * λ)), (((n * a) * λ), (k * λ))], u], k, 1, (n * a)) → se1
  22. RemoveUndefined(LC) → LC2
  23. Length(LC) → tir
  24. Length(LC2) → tirc
  25. PolyLine(K + Append[(0, 0), Sequence[(HF * Element[LR, j]), j, 1, Length[LR]]]) → LR2
  26. Vector(A, B + (2, 0)) → v
  27. Vector(A, D + (0, 2)) → w
  28. Centroid(Polígon2) → CP
  29. RemoveUndefined(LI) → LI2
  30. Segment(K + (0, (h1 * S1 / a^(2))), K + ((h2 * (Length[LR] - 1)), (h1 * S1 / a^(2)))) → l
  31. PointList(LC2) → LC3

AI Drawing Prompt Reference

This is a geometric figure about the probability of a rectangle's existence. The figure includes a rectangle and a square, showing their side lengths and area relationships to illustrate probability calculations. The construction order starts with the rectangle, then adds the square, and finally uses intersections and transformations to show their relationships. The elements include the rectangle, square, intersection points, and related line segments, presenting a clear representation of geometric probability.
The prompt above was compiled from the original drawing commands and is for reference only.