4-7%20Medians,%20Altitudes,%20and%20Perpendicular%20Bisectors

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4-7 Medians, Altitudes, and Perpendicular
Bisectors
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Median

• of a triangle is a segment from a vertex to the
  midpoint of the opposite side.

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Altitude

• Segment from a vertex perpendicular to the line
  containing the opposite side. In an acute
  triangle all 3 altitudes are inside the triangle.
• Altitudes can be outside the triangle

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Altitudes in right triangles

• Two of the altitudes are the actual legs of the
  triangle.

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Altitudes in an obtuse triangle

• In an obtuse triangle, 2 of the altitudes are
  outside the triangle.

Red lines are altitudes for this triangle.
Notice that some of them are outside of the
triangle, and one is on the inside.
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Perpendicular Bisectors

• A line or ray that is perpendicular to a segment
  at the segments midpoint.

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Theorem

• If a point lies on the perpendicular bisector of
  a segment, then the point is equidistant from the
  endpoints of the segment.

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Theorem

• If a point is equidistant from the endpoints of a
  segment, then the point lies on the ___________
  __________ of the segment.

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Theorem

• If a point lies on the bisector of an angle, then
  the point is equidistant from the sides of the
  angle.
• What kinds of lines do we use to measure distance?

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A
X
Z
P
C
B
Y
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Theorem

• If a point is equidistant from the sides of an
  angle, then the point lies on the bisector of the
  angle.

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Centers of a triangle
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Homework

• Pg. 155 1-9 CE
• Pg. 156 7-12
• Pg. 161 11-15
• Pg. 162 1-8