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Two Rectangles and Triangle's Perimeter or Area

Two Rectangles and Triangle's Perimeter or Area
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2016-01-26 04:26
TrianglePolygonOptimization
In collections RectTri Area vs. Perimeter Drafts
This example demonstrates the relationship between two rectangles and a triangle when they share either the perimeter or the area, exploring how well rectangles enclose area. It involves core math concepts like perimeter, area, geometric transformations, quadratic and logarithmic functions. Constructed using commands such as Polygon, Circle, Point, Segment, Intersect, Distance, Rotate, Angle, Line, OrthogonalLine, Locus, and objects like PRectMover, poly1, triPerim, rootTriArea. Suitable for high school mathematics education, illustrating geometric transformations and function applications.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Circle(PRectMover, halfTriPerim) → c_1
  3. Point(c_1) → RectRot
  4. Segment(PRectMover, RectRot) → a_1
  5. Circle(PRectMover, (0.99 * a_1)) → q_1
  6. Intersect(q_1, a_1) → V,
  7. Segment(V, PRectMover) → PRectDomain
  8. Point(Circle[A, rootTriArea]) → D
  9. Segment(A, D) → e
  10. Polygon(A, D, 4) → poly2, f, g, h, i, E, F
  11. Point(Circle[D, rootTriArea]) → G
  12. Segment(D, G) → l
  13. Point(Circle[G, rootTriArea]) → H
  14. Segment(G, H) → m
  15. Point(Circle[H, rootTriArea]) → I
  16. Segment(H, I) → n
  17. Point(Circle[C, a]) → J
  18. Segment(C, J) → p
  19. Point(Circle[J, c]) → K
  20. Segment(J, K) → q
  21. Circle(B, A) → s
  22. Circle(A, B) → t
  23. Intersect(s, t) → M, N, ,
  24. Distance(L, r) → o
  25. Point(PRectDomain) → PRectSizer
  26. Rotate(PRectMover, -90°, PRectSizer) → P'
  27. Angle(PRectMover, PRectSizer, P') → α
  28. Line(PRectSizer, P') → b_1
  29. Segment(PRectMover, PRectSizer) → d_1
  30. Circle(PRectSizer, halfTriPerim - d_1) → e_1
  31. Intersect(e_1, b_1, 1) → S
  32. Segment(PRectSizer, S) → f_1
  33. OrthogonalLine(S, f_1) → g_1
  34. OrthogonalLine(PRectMover, d_1) → h_1
  35. Intersect(g_1, h_1) → T
  36. Polygon(PRectMover, PRectSizer, S, T) → RectWTriPerim, p_1, r_1, s_1, t_1
  37. Segment(P, Q) → d
  38. Point(Circle[ARectMover, f]) → ARectRot
  39. Segment(ARectMover, ARectRot) → ARectDomain
  40. Point(ARectDomain) → ARectSizer
  41. OrthogonalLine(ARectSizer, ARectDomain) → k
  42. OrthogonalLine(ARectMover, ARectDomain) → i_1
  43. Circle(ARectSizer, equiAreaRectHeight) → k_1
  44. Intersect(k_1, k, 2) → Z
  45. OrthogonalLine(Z, k) → j_1
  46. Intersect(i_1, j_1) → A_1
  47. Polygon(ARectMover, ARectSizer, Z, A_1) → RectWTriArea, u, w, z_1, a_2
  48. Locus(R_A, ARectSizer) → loc1
  49. Locus(R_P, PRectSizer) → loc2

AI Drawing Prompt Reference

Draw a triangle with two rectangles, one sharing perimeter and the other area. Show how the rectangles change as the triangle's vertex moves, highlighting key points, lines, and circles involved in the perimeter or area sharing. Start with the triangle, then construct the perimeter-sharing rectangle using circles and intersections, followed by the area-sharing rectangle, adjusting its size based on area relationships.
The prompt above was compiled from the original drawing commands and is for reference only.