Transcript Medians, Altitudes and Angle Bisectors

Medians, Altitudes and
Angle Bisectors
Every triangle has
1. 3 medians,
2. 3 angle bisectors and
3. 3 altitudes.
B
A
C
Given ABC, identify the opposite side
1. of A.
BC
2. of B.
AC
3. of C.
AB
Any triangle has three medians.
B
L
M
A
N
C
Let L, M and N be the midpoints of AB, BC and AC respectively.
Hence, CL, AM and NB are medians of ABC.
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.
Any triangle has three angle bisectors.
B
E
A
F
MD
C
In the figure, AF, DB and EC are angle bisectors of ABC.
Note: An angle bisector and a median of a triangle are
Definition of ansometimes
Angle Bisector
of a Triangle
different.
A segment
an angleofbisector
of a triangle if and only if
Let
M be theis midpoint
AC.
a) itis lies
in the and
ray which
bisects
angle ofofthe
triangle and
BM
a median
BD is an
angleanbisector
ABC.
b) its endpoints are the vertex of this angle and a point on
the opposite side of that vertex.
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.
B
C
A
ACUTE
OBTUSE
Can a side of a triangle be its altitude? YES!
A
G
C
B
RIGHT
If ABC is a right triangle, identify its altitudes.
BG, AB and BC are its altitudes.
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.
T
V
M
R
S
U
Let TU be a perpendicular bisector of RS.
RT = TS
Then, what can you say about T, V and U?
RV = VS
RU = US
Theorem: If a point lies on the perpendicular
bisector of a segment, then the point is
equidistant from the endpoints of the
segment.
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.
H
F
G
Given: HF = HG
Conclusion: H lies on the perpendicular bisector of FG.
T
V
R
S
If T is equidistant from R and S and similarly, V is
equidistant from R and S, then what can we say about TV?
TV is the perpendicular bisector of RS.
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.
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.
Let AD be a bisector of BAC,
M
B
P lie on AD,
PM  AB at M,
P
A
D
NP  AC at N.
N
C
Then P is equidistant from AB and AC.
Theorem: If a point lies on the bisector of
an angle, then the point is equidistant
from the sides of the angle.
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.