Honors Geometry

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Honors Geometry

• Lesson 5.2
• Special Segments in a Triangle
• Geometry Applet

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What You Should LearnWhy You Should Learn It

• Goal 1 How to identify special segments in a
  triangle
• Goal 2 How to use properties of special segments
  to solve problems in geometry and in real life
• Using the properties of special segments related
  to triangles can help you answer questions about
  triangular objects, such as how to find the
  balancing point of a triangular model

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Perpendicular Bisector of a Triangle

• A perpendicular bisector of a triangle is a
  segment that is part of a perpendicular bisector
  of one of the sides
• It lies in the triangles interior
• The endpoints are points of the triangle
• May or may not have a vertex as one endpoint

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Angle Bisector of a Triangle

• An angle bisector of a triangle is a segment that
  bisects one of the angles of the triangle
• It lies in the triangles interior
• The endpoints are points of the triangle
• A vertex is one of the endpoints

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Median of a Triangle

• A median is a segment whose endpoints are a
  vertex and the midpoint of the opposite side
• It lies in the triangles interior
• The endpoints are points of the triangle
• A vertex is one of the endpoints

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Altitude of a Triangle

• An altitude is a segment from a vertex that is
  perpendicular to the opposite side or to the line
  containing the opposite side
• An altitude may lie inside or outside the
  triangle
• A vertex is one of the endpoints

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Lesson Investigation
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Using Concurrency Properties

• Every triangle has three perpendicular bisectors,
  three angle bisectors, three medians, and three
  altitudes

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Theorem 5.5Concurrency Properties (circumcenter)

• The lines containing the perpendicular bisectors
  of a triangle are concurrent (meeting in one
  point).
• Their common point is the circumcenter of the
  triangle.
• The circumcenter is equidistant from the vertices
  of the triangle

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Theorem 5.5Concurrency Properties (incenter)

• The angle bisectors of a triangle are concurrent.
• Their common point is the incenter of the
  triangle.
• The incenter is equidistant from the three sides
  of the triangle

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Theorem 5.5Concurrency Properties (centroid)

• The medians of a triangle are concurrent.
• Their common point is the centroid of the
  triangle
• The centroid is ? of the distance from each
  vertex to the midpoint of the opposite side

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Theorem 5.5Concurrency Properties (Orthocenter)

• The lines containing the altitudes of a triangle
  are concurrent
• Their common point is the orthocenter of the
  triangle

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Find the balancing point of a triangular model
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THE END