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Exploring Rectangle Area Formula

Exploring Rectangle Area Formula
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2015-03-26 14:43
VectorCoordinate GeometryGeometric Construction
This example demonstrates how the area formula for a rectangle remains valid when the side lengths are not integers. It involves core math concepts like rectangle area formula, vectors, and coordinate geometry. Constructed using GeoGebra commands such as Point, Segment, Circle, OrthogonalLine, Intersect, Div, Sequence, and Polygon. Suitable for high school students to understand the universality of mathematical formulas.

GeoGebra Commands & Parameters

  1. Point(Circle[A, b]) → B
  2. Segment(A, B) → c
  3. Point(Circle[B, d]) → C
  4. Segment(B, C) → e
  5. Circle(B, C) → f
  6. OrthogonalLine(B, c) → g
  7. Intersect(f, g, 2) → D
  8. Segment(D, B) → h
  9. OrthogonalLine(D, h) → i
  10. OrthogonalLine(A, c) → j
  11. Intersect(i, j) → E
  12. Segment(E, D) → k
  13. Segment(E, A) → l
  14. Div(4.2, 1) → m
  15. Div(4.2, 0.5) → n
  16. Sequence(E + ((D - E) / (D - E) / a * k), k, 1, Div[b, 1 / a]) → list1
  17. Sequence(A + ((B - A) / (B - A) / a * k), k, 1, Div[b, 1 / a]) → list2
  18. Sequence((x(A), y(A) + (1 / a * k)), k, 1, Div[d, 1 / a]) → list3
  19. Sequence((x(B), y(B) + (1 / a * k)), k, 1, Div[d, 1 / a]) → list4
  20. Sequence(Segment[Element[list1, k], Element[list2, k]], k, 1, Div[b, 1 / a]) → list5
  21. Sequence(Segment[Element[list3, k], Element[list4, k]], k, 1, Div[d, 1 / a]) → list6
  22. Point(h) → F
  23. Point(c) → G
  24. Point(Circle[H, 1]) → I
  25. Segment(H, I) → p
  26. OrthogonalLine(I, p) → q
  27. Circle(I, H) → r
  28. Intersect(r, q, 1) → J
  29. OrthogonalLine(J, q) → s
  30. OrthogonalLine(H, p) → t
  31. Intersect(s, t) → K
  32. Segment(H, K) → a_1
  33. Segment(K, J) → b_1
  34. Segment(J, I) → c_1
  35. Sequence((x(H) + (1 / a * k), y(H)), k, 1, Div[1, 1 / a]) → list7
  36. Sequence((x(K) + (1 / a * k), y(K)), k, 1, Div[1, 1 / a]) → list8
  37. Sequence((x(K), y(K) + (1 / a * k)), k, 1, Div[1, 1 / a]) → list9
  38. Sequence((x(J), y(J) + (1 / a * k)), k, 1, Div[1, 1 / a]) → list10
  39. Sequence(Segment[Element[list7, k], Element[list8, k]], k, 1, Div[1, 1 / a]) → list11
  40. Sequence(Segment[Element[list9, k], Element[list10, k]], k, 1, Div[1, 1 / a]) → list12
  41. Polygon(H, K, J, I) → poly1, h_1, k_1, j_1, i_1
  42. Polygon(A, W, M, L) → poly2, a_2, w, m_1, l_1
  43. Point(c_1) → N
  44. Point(b_1) → O

AI Drawing Prompt Reference

Draw a rectangle to demonstrate the universality of its area formula with non-integer side lengths. Start by constructing two perpendicular sides of the rectangle. Extend lines from the vertices along these sides at proportional intervals to form a grid. Then, draw the diagonal inside the rectangle and mark equal divisions along it, connecting these points to create a grid. Finally, construct a similar shape outside the rectangle, labeling key points and segments to illustrate the applicability of the area formula.
The prompt above was compiled from the original drawing commands and is for reference only.