Transcript 5.4 Medians and Altitudes

5.4 Medians and Altitudes
Vocabulary…

Concurrent- 3 or more lines, rays, or segments
that intersect at the same point
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Median of a Triangle – a segment from a
vertex to the midpoint of the opposite side
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Centroid – point of concurrency of 3 medians
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Altitude of a Triangle – the perpendicular
segment from a vertex to the opposite side or to
the line that contains the opposite side – there
are 3 in a
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Orthocenter - the point where the 3 altitudes of
a
intersect
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Theorem 5.8: Concurrency of Medians
of a
-the medians of a
intersect at a point
that is 2/3 the distance from each vertex to the
midpoint of the opposite side.
Theorem 5.9: Concurrency of Altitudes
of a
-the lines containing the altitudes of a
are concurrent
In RST, Q is the centroid and SQ = 8. Find QW and
SW.
SQ =
2
SW
3
8 = 2 SW
3
12 = SW
Concurrency of Medians of a
Triangle Theorem
Substitute 8 for SQ.
Multiply each side by the reciprocal, 3 .
2
Then QW = SW – SQ = 12 – 8 = 4.
So, QW = 4 and SW = 12.
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Practice
In Exercises 1–3, use the diagram.
G is the centroid of ABC.
1.
If BG = 9, find BF.
ANSWER 13.5
2.
If BD = 12, find AD.
ANSWER 12
3.
If CD = 27, find GC.
ANSWER 18
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Find the orthocenter.
Find the orthocenter P in an acute, a right, and an
obtuse triangle. (Draw 3 altitudes…drop
perpendicular lines from vertex to opposite side.)
SOLUTION
Acute triangle
Right triangle
Obtuse triangle
P is inside triangle. P is on triangle. P is outside triangle.
Can you answer these??????
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Look at the
and answer the following:
1. Is BD a median of
ABC?
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2. Is BD an altitude
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3. Is BD a perpendicular bisector?
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ABC?
B
B
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A
D
C
A
D
C