Constructing Perpendicular Bisectors

1
Constructing Perpendicular Bisectors

• Each segment has exactly one midpoint.
• A segment bisector is a line, ray, or segment
  that passes through the midpoint.
• A perpendicular bisector is a line, ray, or
  segment that passes through the midpoint and is
  perpendicular to the segment.

2
Constructing Perpendicular Bisectors

• Investigation 1
• Finding the Right Bisector
• Step 1
• Draw segment PQ on the sheet of patty paper
• Step 2
• Fold the patty paper so that endpoint P and Q lie
  on top of each other, or coincide
• Crease your paper along the fold
• Step 3
• Unfold your paper and draw a line along the
  crease
• Make a conjecture about the line you drew
• Check your conjecture with your ruler and
  protractor

3
Constructing Perpendicular Bisectors

• Investigation 1
• Finding the Right Bisector
• Step 4
• Place three points, A, B, and C on the bisector
• Use a compass to compare the distance from each
  point to both endpoints of the segment
• Make a conjecture about points on the line
• C-5  Perpendicular Bisector Conjecture
• If a point is on the perpendicular bisector of a
  segment, then it is equidistant from both
  endpoints

4
Constructing Perpendicular Bisectors

• Investigation 2
• Right Down the Middle
• Step 1
• Draw a line segment. Using one endpoint as
  center, swing an arc on one side of the segment
• Step 2
• Using the same compass setting, but using the
  other endpoint as center, swing a second arc
  intersecting the first

5
Constructing Perpendicular Bisectors

• Investigation 2
• Right Down the Middle
• Step 3
• Repeat steps 1 and 2 on the other side of the
  segment or for a different length arc
• Are the points where arcs of equal radius
  intersect on the perpendicular bisector?
• C-5  Converse of the Perpendicular Bisector
  Conjecture
• If a point is equidistant from the endpoints of a
  segment, then it is on the perpendicular bisector
  of the segment

6
Constructing Perpendicular Bisectors

• Constructing the perpendicular bisector of a
  segment is one way to find its midpoint
• Finding the midpoint of one or more sides of a
  triangle allows you to construct the median and
  midsegment
• A median is a segment connecting the vertex of a
  triangle to the midpoint of the opposite side
• A midsegment is a segment connecting the
  midpoints of two sides of a triangle