Lesson 32: Median, Angle Bisector, Altitude and Perpendicular Bisector

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Lesson 32 Median, Angle Bisector, Altitude and
Perpendicular Bisector

• Nov. 13, 2009

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Homework

• Geometry, P156-158, 7-17, 19, 23

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Do Now

• Suppose that you and Pat are partners in a game
  in which you must locate various clues to win.
  You are told to pick up your next clue at a point
  that
• is as far from the fountain as from the oak tree
• and
• is 10 m from the flag pole.
• Q Can you find a point that meets both
  requirements 1 and 2?

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Question

• What kinds of segments can you draw from a vertex
  to the opposite side in a triangle?

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Vocabulary

• Median of a triangle a segment from a vertex to
  the midpoint of the opposite side.
• Altitude of a triangle the perpendicular segment
  from a vertex to the line that contains the
  opposite side. (Q Is it important to mention the
  line that contains the opposite side?)

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Questions

• What kind of triangle has one angle bisector that
  is also altitude and median?
• What kind of triangle has three angle bisectors
  that are also altitudes and medians?

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Vocabulary

• Perpendicular bisector of a segment a line (or
  ray or segment) that is perpendicular to the
  segment at its midpoint. (Q Does perpendicular
  bisector only exist in a triangle?)

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Equidistance Theorems of Perpendicular Bisectors

• Theorem 4-5 If a point lies on the perpendicular
  bisector of a segment, then the point is
  equidistant from the endpoints of the segment.
• Theorem 4-6 If a point is equidistant from the
  endpoints of a segment, then the point lies on
  the perpendicular bisector of the segment.
• Q Can you combine both theorems into one
  biconditional statement?

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Example

• Given S is equidistant from E and D V is
  equidistant from E and D.
• Prove is the perpendicular bisector of

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Equidistance Theorems of Angle Bisectors

• Theorem 4-7 If a point lies on the bisector of
  an angle, then the point is equidistant from the
  sides of the angle.
• Theorem 4-8 If a point is equidistant from the
  sides of an angle, then the point lies on the
  bisector of the angle.
• Q Can you combine both theorems into one
  biconditional statement?

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Example

• Given bisects ?ADE bisects ?DEC
• Prove bisects ?ABC