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Trapezoid Area and Midsegment – Dynamic Proof of the Formula

Trapezoid Area and Midsegment – Dynamic Proof of the Formula
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2012-10-26 18:21
PolygonCoordinate GeometryGeometric Construction
This dynamic GeoGebra example illustrates how the midsegment of a trapezoid can be used to prove the area formula \(A = \frac{(a+b)h}{2}\). It covers key concepts such as the properties of trapezoids, the midsegment theorem, and dynamic geometry proof techniques. The construction employs GeoGebra commands like Line, Point, OrthogonalLine, Polygon, Midpoint, Segment, Intersect, If, and Rotate to create an interactive trapezoid with adjustable bases a, b and height h via sliders. Core objects include vertices A, B, C, D, midpoints E, F, etc. The material is designed for high‑school geometry and introductory college mathematics courses, providing an intuitive, visual proof for teaching or self‑study.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(C, a) → b
  3. Point(a) → D
  4. Point(a) → E
  5. Point(b) → G
  6. OrthogonalLine(D, b) → c
  7. OrthogonalLine(G, a) → h
  8. Polygon(D, E, G, C) → mnohoúhelník1, d, e, g, c_1
  9. Midpoint(c_1) → F
  10. Midpoint(e) → H
  11. Segment(F, H) → f
  12. OrthogonalLine(F, b) → i
  13. OrthogonalLine(H, b) → j
  14. Intersect(j, a) → K
  15. Intersect(i, a) → L
  16. Polygon(D, L, F) → mnohoúhelník2, f_1, d_1, l
  17. Polygon(H, K, E) → mnohoúhelník3, e_1, h_1, k
  18. Polygon(L, K, H, G, C, F) → mnohoúhelník4, l_1, k_1, h_2, g_1, c_2, f_2
  19. If(m < 0.5, 0, If[m < 1.5, m - 0.5, 1]) → alfa
  20. Rotate(D, -((alfa * 180)°), F) → D'
  21. Rotate(L, -((alfa * 180)°), F) → L'
  22. Rotate(F, -((alfa * 180)°), F) → F'
  23. Polygon(D', L', F') → mnohoúhelník2', f', d', l'
  24. Rotate(H, (alfa * 180)°, H) → H'
  25. Rotate(K, (alfa * 180)°, H) → K'
  26. Rotate(E, (alfa * 180)°, H) → E'
  27. Polygon(H', K', E') → mnohoúhelník3', e', h', k'
  28. OrthogonalLine(C, d) → n
  29. Intersect(n, a) → I
  30. Segment(C, I) → p