5-2 Medians and Altitudes of Triangles

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5-2 Medians and Altitudes of Triangles
You identified and used perpendicular and angle
bisectors in triangles.

• Identify and use medians in triangles.

• Identify and use altitudes in triangles.

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Median

• A median of a triangle is a segment whose
  endpoints are a vertex and the midpoint of the
  opposite side.
• (Goes from vertex to opposite side)

Median
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• Every triangle has three medians that are
  concurrent.
• The point of concurrency of the medians of a
  triangle is called the centroid and is always
  inside the triangle.

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Balancing Act

• Balance your triangle on the eraser end of your
  pencil. Mark the point. This is the centroid of
  the triangle.
• Fold your triangle to find the midpoint of each
  side of your triangle Connect each vertex to the
  midpoint of the opposite side.
• Do all three line segments meet at one point?

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Centroid

• The point of concurrency of the medians of any
  triangle is called the centroid.
• The centroid is the center of balance (or center
  of gravity) of the triangle.

Centroid
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In ?XYZ, P is the centroid and YV 12. Find YP
and PV.
Centroid Theorem
YV 12
Simplify.
YP PV YV Segment Addition 8 PV 12 YP
8 PV 4 Subtract 8 from each side.
Answer YP 8 PV 4
 
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In ?LNP, R is the centroid and LO 30. Find LR
and RO.
A. LR 15 RO 15 B. LR 20 RO 10 C. LR
17 RO 13 D. LR 18 RO 12
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In ?JLN, JP 16. Find PM.
A. 4 B. 6 C. 16 D. 8
 
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Altitude (height)

• An altitude of a triangle is a perpendicular
  segment drawn from a vertex to the line that
  contains the opposite side.
• (Vertex to opposite side at a right angle.)

Altitude
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• Every triangle has three altitudes. If extended,
  the latitudes of a triangle intersect in a common
  point called the orthocenter.

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COORDINATE GEOMETRY The vertices of ?HIJ are
H(1, 2), I(3, 3), and J(5, 1). Find the
coordinates of the orthocenter of ?HIJ.
Point-slope form
Distributive Property
Add 1 to each side.
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Point-slope form
Distributive Property
Subtract 3 from each side.
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Then, solve a system of equations to find the
point of intersection of the altitudes.
Equation of altitude from J
Multiply each side by 5.
Add 105 to each side.
Add 4x to each side.
Divide each side by 26.
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5-2 Assignment

• Page 340, 5-11, 16-19, 27-30