54 Use Medians and Altitudes

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5-4 Use Medians and Altitudes

• Goals
• You will use medians and altitudes of triangles.

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Median

• A median of a triangle is a line which connects a
  vertex with the midpoint of the opposite side.

Midpoint
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Centroids

• The intersection of all three medians is called
  the centroid.
• 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.

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Example 1

• In ?TUV, Y is the centroid. If YU 9, find ZY
  and ZU.

U
W
X
Y
T
Z
V
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Altitude

• An altitude is a line segment that extends from
  one vertex to the opposite side, or an extension
  of that opposite side, and is perpendicular to
  it.
• The intersection of all three altitudes of a
  triangle is known as an orthocenter.

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Orthocenters

• Theorem 5-9
• The lines containing the altitudes of a triangle
  are concurrent.

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Example 4

• Point D is the centroid of ?ABC. Find the value
  of x if BD 4x 5 and BF 9x.

B
E
G
D
A
C
F