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Diagonal of a Rectangular Prism: Build It via Right Triangles and Vectors

Diagonal of a Rectangular Prism: Build It via Right Triangles and Vectors
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2023-12-01 20:14
TriangleAngleVectorSolid GeometryGeometric Construction
In collections Open Middle: Diagonal of a Rectangular Prism
This example demonstrates how to construct the space diagonal of a rectangular prism between opposite vertices, and how to obtain its length by organizing the geometry into right-triangle relationships. Key concepts include solid geometry (rectangular prism, segment/diagonal), the Pythagorean theorem for right triangles, vector ideas for spatial distance, and translating 3D length into computable segment measures through auxiliary constructs. Construction highlights: define key vertices and edges using Point and Segment; enforce right-angle structures using Angle and OrthogonalLine; use Sphere/Circle together with Rotate and Translate to support the spatial visualization and positioning; compute auxiliary points via Intersect; and form the relevant triangles/regions with Polygon while tracking the diagonal segment for measurement. Suitable for advanced middle school, high school geometry, and early undergraduate instruction focused on spatial reasoning, modeling, and revisiting Pythagorean relationships in 3D.

GeoGebra Commands & Parameters

  1. Textfield(a) → InputBox1
  2. Textfield(b) → InputBox2
  3. Textfield(c) → InputBox3
  4. Textfield(d) → InputBox4
  5. Textfield(f) → InputBox5
  6. Textfield(g) → InputBox6
  7. If(Length[Mode[l1]] ≟ 0, "No digits repeat!", "Repeated digits exist.") → text4
  8. Textfield(h) → InputBox7
  9. PointIn(Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → D
  10. Circle(D, (10 * a) + b, zAxis) → e
  11. Point(e) → E
  12. Line(D, E) → i
  13. Rotate(D, ((-pi)) / 2, E, Plane[(0, 0, 0), (1, 0, 0), (1, 1, 0)]) → D'
  14. Line(D', E) → j
  15. Segment(D, E) → k
  16. Circle(E, d, Line[E, E + (0, 0, 1)]) → p
  17. Intersect(j, p) → F, H
  18. Segment(E, H) → l
  19. Angle(H, E, D) → α
  20. Segment(D, H) → m
  21. Line(H, H + (0, 0, 1)) → n
  22. Sphere(H, c) → o
  23. Intersect(n, o) → I, J
  24. Segment(H, J) → q
  25. Segment(J, D) → r
  26. Polygon(D, E, H) → t1, h_1, d_1, e_1
  27. Polygon(D, H, J) → t2, j_1, d_2, h_2
  28. OrthogonalLine(J, Vector[J, D], Vector[J, H]) → s
  29. Line(H, J) → t
  30. Circle(J, sqrt((100 * f) + (10 * g) + h), s) → c_1
  31. Point(c_1) → K
  32. Segment(J, K) → f_1
  33. Line(E, E + (0, 0, 1)) → g_1
  34. Sphere(E, Radius[c_1]) → u
  35. Sphere(E, c) → v
  36. Intersect(g_1, v) → L, M
  37. Line(D, d_1) → i_1
  38. Sphere(D, d) → w
  39. Intersect(i_1, w) → N, O
  40. Translate(O, Vector[E, M]) → O'
  41. Translate(D, Vector[E, M]) → D'_{1}
  42. Polygon(D, O, O', D'_{1}) → q1, d_3, o_1, o', d'_{1}
  43. Polygon(D'_{1}, M, E, D) → q2, k_1, m_1, e_2, d_4
  44. Polygon(M, J, H, E) → q3, m_2, j_2, h_3, e_3
  45. Polygon(H, J, O', O) → q4, h_4, j_3, l_1, o_2
  46. Polygon(D'_{1}, M, J, O') → q5, n_1, m_3, j_4, p_1
  47. Segment(O', D'_{1}) → q_1
  48. Segment(D'_{1}, D) → r_1
  49. Segment(D, O) → s_1
  50. Segment(O, O') → t_1
  51. Segment(E, M) → a_1
  52. Segment(M, D'_{1}) → b_1
  53. Segment(M, J) → f_2
  54. Segment(J, O') → g_2
  55. Segment(H, O) → i_2
  56. Angle(J, H, D) → β

AI Drawing Prompt Reference

Draw a rectangular prism and highlight its space diagonal. The prism is formed by three mutually perpendicular planes, and the diagonal connects two opposite vertices. Auxiliary points, line segments, circles, and spheres are used to illustrate the calculation of the diagonal's length and spatial relationships. The construction order is: first establish the base plane and points, then add auxiliary circles and points, then connect line segments and angles, and finally form the diagonal region.
The prompt above was compiled from the original drawing commands and is for reference only.