Medians, Altitudes and Angle Bisectors

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Medians, Altitudes and Angle Bisectors
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• Every triangle has
• 1. 3 medians,
• 2. 3 angle bisectors and
• 3. 3 altitudes.

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B
A
C

• Given ?ABC, identify the opposite side
• of A.
• of B.
• of C.

4
Any triangle has three medians.
B
L
M
A
N
C
Definition of a Median of a Triangle A median of
a triangle is a segment whose endpoints are a
vertex of a triangle and a midpoint of the side
opposite that vertex.
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Any triangle has three angle bisectors.
B
E
F
A
C
D
M
Note An angle bisector and a median of a
triangle are sometimes different.
Definition of an Angle Bisector of a Triangle A
segment is an angle bisector of a triangle if and
only if a) it lies in the ray which bisects an
angle of the triangle and b) its endpoints are
the vertex of this angle and a point on the
opposite side of that vertex.
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Any triangle has three altitudes.
Definition of an Altitude of a Triangle A
segment is an altitude of a triangle if and only
if it has one endpoint at a vertex of a triangle
and the other on the line that contains the side
opposite that vertex so that the segment is
perpendicular to this line.
ACUTE
OBTUSE
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Can a side of a triangle be its altitude?
YES!
A
G
C
B
RIGHT
If ?ABC is a right triangle, identify its
altitudes.
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D
B
C
If BD DC, then we say that
D is equidistant from B and C.

• Definition of an Equidistant Point
• A point D is equidistant from B and C if and
  only if BD DC.

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T
V
M
R
S
U
RT TS RV VS RU US
Then, what can you say about T, V and U?

• Theorem If a point lies on the perpendicular
  bisector of a segment, then the point is
  equidistant from the endpoints of the segment.

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• Theorem If a point lies on the perpendicular
  bisector of a segment, then the point is
  equidistant from the endpoints of the segment.

The converse of this theorem is also true
Theorem If a point is equidistant from the
endpoints of a segment, then the point lies on
the perpendicular bisector of the segment.
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H
F
G
Given HF HG Conclusion H lies on the
perpendicular bisector of FG.
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T
V
R
S
Theorem If two points and a segment lie on the
same plane and each of the two points are
equidistant from the endpoints of the segment,
then the line joining the points is the
perpendicular bisector of the segment.
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• Definition of a Distance Between a Line and a
  Point not on the Line
• The distance between a line and a point not on
  the line is the length of the perpendicular
  segment from the point to the line.

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B
Let AD be a bisector of ?BAC, P lie on AD, PM ?
AB at M, NP ? AC at N.
M
P
A
N
C

• Theorem If a point lies on the bisector of an
  angle, then the point is equidistant from the
  sides of the angle.

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Theorem If a point lies on the bisector of an
angle, then the point is equidistant from the
sides of the angle.
The converse of this theorem is not always true.
Theorem If a point is in the interior of an
angle and is equidistant from the sides of the
angle, then the point lies on the bisector of the
angle.