Medians and Altitudes

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Section 3-8

• Medians and Altitudes

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A median of a triangle is a segment whose
endpoints are a vertex of the triangle and
the midpoint of the opposite side.
D
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The three medians of a triangle are concurrent.
The point of concurrency is called the centroid
of the triangle. The centroid is always inside
the triangle.
The medians of a triangle have a special
concurrency property.
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Concurrency of Medians of a Triangle
CENTROID
The medians of a triangle intersect at a point
that is two-thirdsof the distance from each
vertex to the midpoint of the opposite side.
If P is the centroid of ?ABC, then
P
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The centroid of a triangle can be used as its
balancing point.
A triangular model of uniformthickness and
density willbalance at the centroid of the
triangle.
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SOLUTION
So, RP 10 and RT 15.
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An altitude of a triangle is the perpendicular
segment from a vertex to the opposite side or to
the line that contains the opposite side. An
altitude can lie inside, on, or outside the
triangle.
Every triangle has three altitudes. The lines
containing the altitudes are concurrent and
intersect at a point called the orthocenter of
the triangle.
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SOLUTION
Draw an example.
The three altitudes intersect at G, a point
inside the triangle.
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SOLUTION
This implies that the orthocenter is on the
triangle at M, the vertex of the right angle of
the triangle.
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SOLUTION
The three lines that contain the altitudes
intersect at W, a point outside the triangle.
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Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle
are concurrent.
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PO 11
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