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Mastering Perpendicular Line Constructions: Step-by-Step Guide

Explore the art of line constructions using compass and straightedge, focusing on perpendicular lines. Learn three key constructions for perpendicular lines with detailed steps and examples. Enhance your geometry skills with practical knowledge and interactive exercises.

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Mastering Perpendicular Line Constructions: Step-by-Step Guide

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  1. Perpendicular Line ConstructionsLindsay ChenowethJonathan Bojarski

  2. Introduction • What is a construction? • A “drawing” using only a compass and straightedge that renders “exact” results. • Construction is not simply drawing, it is like building a drawing. • No instrument or surface can actually create exact results. • Why do we need constructions?

  3. Perpendicular Lines • Perpendicular lines: two lines that intersect, and at their intersection form four angles that equal 90 degrees each.

  4. Perpendicular Lines • 3 Constructions of Perpendicular Lines: • Given two points A and B on a line, construct any line perpendicular to the given line. • Given a line and a point A on that line, construct a perpendicular line through point A. • Given a line and a point A off of that line, construct a line from point A perpendicular to the given line.

  5. Construction 1 • Given two points on a line, construct any line perpendicular to the given line. B A

  6. Construction 1 • Step 1: Draw a circle centered at point A.

  7. Construction 1 • Step 2: Draw a circle with the same radius centered at point B.

  8. Construction 1 • Step 3: Draw points at the places where the two circles intersect.

  9. Construction 1 • Step 4: Draw a line that connects the two points. This line should be perpendicular to line AB.

  10. Construction 2 • Given a line and a point A on that line, construct a line perpendicular to that line through A.

  11. Construction 2 • Step 1: Draw a circle of any radius centered at point A.

  12. Construction 2 • Step 2: Draw points at both places where the circle intersects the given line. Name these points B and C.

  13. Construction 2 • Step 3: Draw a circle centered at point B with a radius larger than segment BA. Draw a circle centered at point C with the same radius as circle B.

  14. Construction 2 • Step 4: Draw points at both places where circles B and C intersect. Connect these points with a line. This line should be perpendicular to line BAC through point A.

  15. Construction 3 • Given a line and a point A off of that line, construct a line perpendicular to the given line that runs through point A.

  16. Construction 3 • Step 1: Draw a circle centered at point A that intersects the given line in two places. Draw points where the circle intersects the line. Name these points B and C.

  17. Construction 3 • Step 2: Draw a circle centered at point B with a radius greater than or equal to BA. Draw a circle centered at point C with a radius of CA.

  18. Construction 3 • Step 3: Draw points where these circles intersect. Draw a line that connects this point with point A. This line should be perpendicular to line BC.

  19. Constructions Online http://www.mathsnet.net/campus/construction/i_index.html A3: construction 1 B2: construction 2 B4: construction 3

  20. Question • Using what you know about the different ways to construct perpendicular lines, can you come up with a way to construct parallel lines?

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