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Shapes with Same Area

Shapes with Same Area
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2018-01-24 01:05
Coordinate GeometryCalculusGeometric Construction
This example demonstrates how different geometric shapes can have the same area. It involves core math concepts like area calculation, statistical characteristics, line segments, Pythagorean theorem, integral calculus, and arithmetic mean. The construction uses commands such as Line, Intersect, Point, Join, Segment, Angle, and Rotate from GeoGebra. It is suitable for high school students studying geometry and calculus to understand the relationship between area and shape.

GeoGebra Commands & Parameters

  1. Line(A, B) → h
  2. Line(D, h) → i
  3. Line(A, D) → f
  4. Line(B, f) → g
  5. Intersect(i, g) → C
  6. Point(GridSpaceQ) → Q
  7. Join(Sequence[Sequence[(i, j), i, 34, 74, 1], j, 12, 32, 1]) → GridSpace
  8. Point(GridSpace) → P
  9. Line(P, Q) → j
  10. Point(GridSpace) → S
  11. Line(S, j) → k
  12. Line(P, S) → l
  13. Line(Q, l) → m
  14. Intersect(m, k) → R
  15. Segment(S, R) → RS
  16. Segment(C, D) → CD
  17. Segment(A, B) → AB
  18. Segment(B, C) → BC
  19. Segment(D, A) → DA
  20. Segment(Q, P) → PQ
  21. Segment(R, Q) → QR
  22. Segment(P, S) → SP
  23. Point(GridSpace) → P2
  24. Point(GridSpace) → Q2
  25. Segment(A2, B2) → AB2
  26. Segment(B2, C2) → BC2
  27. Segment(C2, A2) → CA2
  28. Segment(P2, Q2) → PQ2
  29. Point(GridSpaceR) → R2
  30. Segment(Q2, R2) → QR2
  31. Segment(R2, P2) → RP2
  32. Intersect(OrthogonalLine[A2, Line[B2, C2]], Line[B2, C2]) → H2Tri
  33. Segment(H2Tri, H1Tri) → HeightDisp1Tri
  34. Intersect(OrthogonalLine[P2, Line[Q2, R2]], Line[Q2, R2]) → H4Tri
  35. Segment(H4Tri, H3Tri) → HeightDisp2Tri
  36. Intersect(OrthogonalLine[B, Line[C, D]], Line[C, D]) → H2Quad
  37. Intersect(OrthogonalLine[Q, Line[R, S]], Line[R, S]) → H4Quad
  38. Segment(H2Quad, H1Quad) → HeightDisp1Quad
  39. Segment(H4Quad, H3Quad) → HeightDisp2Quad
  40. Segment(H4Tri, Q2) → HeightDispHoriz2Tri
  41. Segment(H2Tri, B2) → HeightDispHoriz1Tri
  42. Segment(H4Quad, S) → HeightDispHoriz2Quad
  43. Segment(H2Quad, C) → HeightDispHoriz1Quad
  44. Angle(A2, H2Tri, C2) → Ang1Tri
  45. Angle(P2, H4Tri, R2) → Ang2Tri
  46. Angle(B, H2Quad, C) → Ang1Quad
  47. Angle(Q, H4Quad, R) → Ang2Quad
  48. Rotate(A2, 2, B2) → A2'
  49. Angle(C2, B2, A2) → AngABCRtTri
  50. Angle(A, B, C) → AngABCRtQuad
  51. Angle(A2, C2, B2) → AngACBRtTri
  52. Angle(R2, Q2, P2) → AngPQRRtTri
  53. Angle(Q2, R2, P2) → AngPRQRtTri
  54. Angle(P, Q, R) → AngPQRRtQuad

AI Drawing Prompt Reference

Draw a quadrilateral ABCD and a triangle PQR with the same area. Start by drawing the quadrilateral ABCD, then draw line segments PQ and RS connecting the diagonals of the quadrilateral. Next, draw triangle PQR to ensure its area matches the quadrilateral. Finally, use rotations and calculations to ensure the heights and angles of the triangle and quadrilateral are correct, forming shapes with equal areas.
The prompt above was compiled from the original drawing commands and is for reference only.