Chapter 5.3 Concurrent Lines, Medians, and Altitudes

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Chapter 5.3 Concurrent Lines, Medians, and
Altitudes
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Vocabularies
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Vocabularies

• Concurrent (line) three or more lines intersect
  in one point
• Point of concurrency the point the concurrent
  lines intersect
• Example

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Vocabularies

• Circumcenter of the triangle The point of
  concurrency of the perpendicular bisectors of a
  triangle
• Example

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Vocabularies

• Circumscribed about A circle is circumscribed
  about a polygon if the vertices of the polygon
  are on the circle. A polygon is circumscribed
  about a circle if all the sides of the polygon
  are tangent to the circle
• Examples

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Vocabularies

• Incenter of the triangle The incenter of a
  triangle is the point of concurrency of the angle
  bisectors of the triangle
• Example

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Vocabularies

• Inscribed in A circle is inscribed in a polygon
  if the sides of the polygon are tangent to the
  circle. A polygon is inscribed in a circle if
  the vertices of the polygon are on the circle
• Examples

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Vocabularies

• Median of a triangle a segment whose endpoints
  are a vertex and the midpoint of the opposite
  side
• Examples

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Vocabularies

• Centroid the point of intersection of the
  medians of that triangle
• Example

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Vocabularies

• Altitude of a triangle the perpendicular
  segment from a vertex to the line containing the
  opposite side. Can be inside or outside the
  triangle.
• Examples
• Acute Triangle Right Triangle Obtuse Triangle

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Vocabularies

• Orthocenter of the triangle the point of
  intersection of the lines containing the
  altitudes of the triangle.
• Example

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Theorems
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Theorem 5.6

• The perpendicular bisectors of the sides of a
  triangle are concurrent at a point equidistant
  from the vertices
• Example

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Theorem 5.7

• The bisectors of the angles of a triangle are
  concurrent at a point equidistant from the sides
• Example

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Theorem 5.8

• The medians of a triangle are concurrent at a
  point that is two thirds the distance from each
  vertex to the midpoint of the opposite side
• Example

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Theorem 5.9

• The lines that contain the altitudes of a
  triangle are concurrent

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Classwork/Homework