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Construct a Segment One-Quarter the Length of PQ

Construct a Segment One-Quarter the Length of PQ
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2015-02-18 02:06
CircleCoordinate GeometryGeometric Construction
In collections PARCC Math III Unit 5 Lesson 1
This example demonstrates how to construct a segment whose length is one-quarter of line segment PQ. Key mathematical concepts include proportion and geometric construction. Commands used include Polygon, Segment, OrthogonalLine, Circle, Intersect, etc. Suitable for high school mathematics education.

GeoGebra Commands & Parameters

  1. Polygon((22, y(Corner[1])), Corner[2], Corner[3], (22, y(Corner[4]))) → mask, a, b, c, d
  2. Segment(P, Q) → PQ
  3. OrthogonalLine(Q, PQ) → lineQ
  4. OrthogonalLine(P, PQ) → LineP
  5. Circle(P, Q) → Circ1
  6. Circle(Q, P) → Circ2
  7. Intersect(Circ1, LineP) → Arc1a, Arc1b
  8. Intersect(Circ2, lineQ) → Arc2a, Arc2b
  9. CircumcircleArc(Arc1a, Q, Arc1b) → Arc1
  10. CircumcircleArc(Arc2a, P, Arc2b) → Arc2
  11. Intersect(Arc2, Arc1) → Mid1a, Mid1b, ,
  12. Segment(Mid1a, Mid1b) → BisectPQ
  13. Intersect(PQ, BisectPQ) → M
  14. Circle(P, M) → Circ3
  15. Circle(M, P) → Circ4
  16. Intersect(Circ4, BisectPQ) → Arc4a, Ark4b
  17. Intersect(Circ3, LineP) → Arc3a, Arc3b
  18. CircumcircleArc(Arc4a, P, Ark4b) → Arc4
  19. CircumcircleArc(Arc3a, M, Arc3b) → Arc3
  20. Intersect(Arc3, Arc4) → Mid2b, Mid2a, ,
  21. Segment(Mid2a, Mid2b) → BisectPM
  22. Intersect(PQ, BisectPM) → N
  23. Segment(P, N) → PN
  24. Segment(P, M) → PM

AI Drawing Prompt Reference

Draw line segment PQ, then construct the perpendicular bisector of PQ to find its midpoint M. Next, draw a circle with center P and radius PM, and another circle with center M and radius PM. The intersections of these circles with the perpendicular bisector of PQ are points N and P. Draw segment PN, which will be one-quarter the length of PQ.
The prompt above was compiled from the original drawing commands and is for reference only.