Constructing Perpendiculars to a Line

1
Constructing Perpendiculars to a Line

• How would you measure the distance from your seat
  to the whiteboard?
• What is the shortest distance from a point (your
  seat) to a line (the edge of the whiteboard)
• Finding a perpendicular from a point to a line is
  one of Euclids constructions with many
  applications
• Its not possible to bisect a line, but you can
  find the perpendicular from a line to any point

2
Constructing Perpendiculars to a Line

• Investigation 1
• Finding the Right Line
• Step 1
• Draw a line and a point P not on the line, as
  shown
• Step 2
• Use your compass to place points A and B on the
  line
• Step 3
• How is PA related to PB? What does this mean
  about point P?
• Step 4
• Construct the perpendicular bisector of segment
  AB and label the midpoint M

3
Constructing Perpendiculars to a Line

• Investigation 1
• Finding the Right Line
• Step 5
• Segment MP is perpendicular to line AB
• Place three points, Q, R, and S, on line AB
• Which point, M, Q, R, or S, is closest to P?
• C-7 Shortest Distance Conjecture
• The shortest distance from a point to a line is
  measured along the perpendicular from the point
  to the line.

4
Constructing Perpendiculars to a Line

• The distance from a point to a line is the length
  of the perpendicular segment from the point to
  the line
• Construction of a perpendicular segment is used
  to locate the altitude of a triangle
• The altitude of a triangle is a perpendicular
  segment from the vertex of a triangle to a line
  containing the opposite side
• The length of the altitude is the height of the
  triangle