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Perpendicular Lines: Definition, Slope Relationship, and Construction

Perpendicular Lines: Definition, Slope Relationship, and Construction
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2015-09-14 20:47
PolygonAngleCoordinate GeometryGeometric Construction
In collections Geometry: Introductory Definitions, Postulates, Theorems
This interactive GeoGebra example demonstrates perpendicular lines by definition and explores consequences such as right angles, slope relationships, and coordinate calculations. It covers perpendicularity, angle measurement, slope, intercepts, fraction simplification, distance, and polygon construction. The construction uses OrthogonalLine to create perpendicular lines, Line, Segment, and Point to define objects, Intersect to locate intersections, Angle to mark right angles, Slope and FractionText to display slope as fractions, GCD and If to reduce numerator and denominator, and Polygon and Distance to support geometric verification. Suitable for middle and high school geometry or coordinate geometry lessons, demonstrations, and student exploration.

GeoGebra Commands & Parameters

  1. GCD(deltaY, deltaX) → GCDm
  2. GCD(numerYint, deltaX) → GCDy
  3. If(sgn(deltaY / deltaX) ≥ 0, "", "-") → textSignN
  4. If(sgn(numerYint / deltaX) > 0, "+", "-") → textSign
  5. If(deltaXR ≟ 1, textSignN + deltaYR, textSignN + "\, \frac{" + deltaYR + "}{" + deltaXR + "}") → textslopeN
  6. If(denomYintR ≟ 1, textSign + numerYintR, textSign + "\frac{" + numerYintR + "}{" + denomYintR + "}") → textYint
  7. If(denomYintR ≟ 1, If[numerYintR ≟ 1, textSign, textSign + numerYintR], textSign + "\frac{" + numerYintR + "}{" + denomYintR + "}") → textMID
  8. OrthogonalLine(B, xAxis) → a_2
  9. OrthogonalLine(A, yAxis) → c
  10. Intersect(a_2, c) → C_1
  11. Line(A, B) → d
  12. Polygon(A, B, C_1) → poly1, c_1, a_1, b_1
  13. Distance(B, C_1) → distanceBC
  14. Distance(A, C_1) → distanceAC
  15. Intersect(a_2, c) → D
  16. Segment(A, D) → e
  17. Segment(C_1, B) → f
  18. GCD(deltaY_1, deltaX_1) → GCDm_1
  19. If(sgn(deltaY_1 / deltaX_1) ≥ 0, "", "-") → textSignN_1
  20. If(deltaXR_1 ≟ 1, textSignN_1 + deltaYR_1, textSignN_1 + "\, \frac{" + deltaYR_1 + "}{" + deltaXR_1 + "}") → textslopeN_1
  21. Line(A_1, B_1) → d_1
  22. OrthogonalLine(A, d) → g
  23. Point(g) → C
  24. Slope(g) → h
  25. FractionText(h) → text4
  26. GCD(deltaY_2, deltaX_2) → GCDm_2
  27. If(sgn(deltaY_2 / deltaX_2) ≥ 0, "", "-") → textSignN_2
  28. If(deltaXR_2 ≟ 1, textSignN_2 + deltaYR_2, textSignN_2 + "\, \frac{" + deltaYR_2 + "}{" + deltaXR_2 + "}") → textslopeN_2
  29. Line(A_2, B_2) → d_2
  30. Angle(B, A, C) → α
  31. Slope(d) → k
  32. FractionText(k) → text8