5-3: Medians and Altitudes

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5-3 Medians and Altitudes
(p. 10)
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5-3 Medians and Altitudes
Objectives

• Apply properties of medians of a triangle.
• Apply properties of altitudes of a triangle.

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5-3 Medians and Altitudes

• Medians of triangles
• Endpoints are a vertex
• and midpoint of opposite side.
• Intersect at a point called the centroid
• Its coordinates are the average of the 3
  vertices.
• The centroid is ? of the distance from each
  vertex to the midpoint of the opposite side.
• The centroid is always located inside the
  triangle.

P
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5-3 Medians and Altitudes
Example 1
S is the centroid of ?LMN. RL 21 and SQ 4.
Find LS and NQ.
Centroid Thm.
Substitute 21 for RL.
LS 14
Simplify.
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Centroid Thm.
NS SQ NQ
Seg. Add. Post.
Substitute 4 for SQ.
Multiply both sides by 3.
12 NQ
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5-3 Medians and Altitudes
Example 2 Problem-Solving Application
A sculptor is shaping a triangular piece of iron
that will balance on the point of a cone. At what
coordinates will the triangular region balance?
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5-3 Medians and Altitudes

• Altitudes of a triangle
• A perpendicular segment from a vertex to the line
  containing the opposite side.
• Intersect at a point called the orthocenter.
• An altitude can be inside, outside, or on the
  triangle.

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5-3 Medians and Altitudes
Example 3
Find the orthocenter of ?XYZ with vertices X(3,
2), Y(3, 6), and Z(7, 2).
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5-3 Medians and Altitudes
Assignment