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Common Aspect Ratio of Five Similar Rectangles (Puzzle)

Common Aspect Ratio of Five Similar Rectangles (Puzzle)
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2020-08-11 08:09
PolygonTransformationOptimizationGeometric Construction
This GeoGebra puzzle demonstrates how to determine the shared aspect ratio among five similar rectangles by enforcing geometric constraints across multiple placements. The key ideas are similarity of rectangles (the invariant aspect ratio), similarity ratios and their transitivity, and representing proportional relationships using geometric transformations. Construction focuses on generating and aligning the rectangles with transformation commands such as Mirror (reflecting across a line), Rotate (turning to match orientations), Dilate (scaling by a similarity factor), and Translate (shifting for proper layout). Straight edges are created using Line and Segment, while the rectangle outlines are assembled with Polygon. Additional objects and variables (e.g., length parameters like l1–l6 and r1–r4, plus parameters such as q1) encode the constraints, and Min can be used to select/track relevant extremal geometric quantities needed for the consistent ratio. Suitable for high-school geometry enrichment, contest preparation, and lessons on how similarity leads to invariant ratios under multiple geometric constraints.

GeoGebra Commands & Parameters

  1. Mirror(A, xAxis) → A'
  2. Min(1, Max[0, anim]) → c
  3. Mirror(A, yAxis) → B
  4. Mirror(A', yAxis) → B'
  5. Polygon(A, A', B', B) → q1, a_1, a', b', b
  6. Dilate({q1}, r1^(c)) → l1
  7. Min(1, Max[0, anim - 1.5]) → d
  8. Translate(Rotate[l1, (d * 90°)], ((d * (a - r1)), 0)) → l2
  9. Min(1, Max[0, anim - 3]) → e
  10. Segment(B', C) → f
  11. Dilate({q1}, r2^(c)) → l3
  12. Dilate(A', r2^(c)) → A''
  13. Translate(l3, (e * (C - A''))) → l3'
  14. Segment(B, D) → g
  15. Dilate({q1}, r3^(c)) → l4
  16. Min(1, Max[0, anim - 6]) → h
  17. Rotate(Dilate[A, r3^(c)], 90°) → E
  18. Translate(Rotate[l4, (h * 90°)], (h * (B - E))) → l5
  19. Min(1, Max[0, anim - 4.5]) → i
  20. Dilate({q1}, r4^(c)) → l6
  21. Dilate(A, r4^(c)) → A_2
  22. Translate(l6, (i * (F - A_2))) → l6'
  23. Line(A, O) → j
  24. Mirror(j, yAxis) → j'

AI Drawing Prompt Reference

Draw five similar rectangles with the same aspect ratio. Start by generating the sides of the rectangles using mirroring and rotation, then adjust their positions and sizes with scaling and translation commands to ensure the same aspect ratio. Finally, use line and polygon commands to draw the rectangle outlines.
The prompt above was compiled from the original drawing commands and is for reference only.