Median of a triangle

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Median of a triangle

• The median of a triangle is a segment whose
  endpoints are a vertex of the triangle and the
  midpoint of the opposite side
• Every triangle has 1 median for each vertex, or 3
  medians total

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Centroid of a triangle

• The centroid of a triangle is the point of
  concurrency of the 3 medians of a triangle.
• This point is also called the center of gravity
• When 3 or more lines intersect at 1 point, they
  are called concurrent lines

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Centroid theoremtheorem 32-1

• The centroid of a triangle is located 2/3 of the
  distance from each vertex to the midpoint of the
  opposite side
• CP 2/3 CN
• BP 2/3 BM
• AP 2/3 AL

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Using the centroid to find segment lengths

• LA 12, PN 3.1 Find AP
• AP 2/3 AL
• AP 2/3(12)
• AP 8
• Find NC
• CP 2/3CN
• CP PN CN
• 2/3 CN 3.1 CN
• 3.1 1/3 CN
• 9.3 CN

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Finding the centroid on a coordinate plane

• Find the centroid of tri DEF with vertices
  D(-3,5),E(-2,1), and F(-7,3)
• Graph the triangle
• Find the midpoint of each segment -
• midp DF (-5,4), midp DE (-2.5,3)
• Since all 3 medians meet at the same point, the
  intersection of any 2 will give the location of
  the centroid.

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• Find the equation of the line from point E to
• (-5,4) or from (-2,1) and (-5,4)
• Slope 3/-3 -1
• Y mxb so 1 -1(-2) b
• -1 b
  y-x-1
• Find the equation of the line from point F to
• (-2.5,3) or from (-7,3) and (-2.5, 3)
• Slope is 0 so
• 3 -x-1
• 4 -x
• -4 x so centroid is (-4,3)

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Find the centroid of tri RTSwith R(-2,2),
T(2,2), S(1, -2)

• Find the medians
• Find the equations of 2 lines
• Find the coordinate of the centroid

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Altitude of a triangle

• The altitude of a triangle is the perpendicular
  segment from the vertex to the line containing
  the opposite side.
• The orthocenter is the point of concurrency of
  the 3 altitudes of a triangle

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Locating the orthocenter

• Draw an acute, a right and an obtuse triangle
• Sketch the altitudes and find the orthocenter for
  each
• Is the orthocenter always in the interior of a
  triangle?

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Use graph paper

• Sketch the orthocenter of the triangle formed by
  (4,2), (-1,1) and (1,6)

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