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Section 5-3
Concurrent Lines, Medians, and
Altitudes
Triangle Medians
A median of a triangle is a line segment drawn from
any vertex of the triangle to the midpoint of the
opposite side.
B
Question: If I printed
this slide in black and
white, what would be
incorrect about the
figure?
D
F
A
C
E
Median
Bisected Segment
Resulting  Segments
AD
BC
BD  DC
BE
AC
AE  EC
CF
AB
AF  FB
Triangle Medians Theorem
B
F
A
D
G
E
2
BG  BE
3
The medians of a
triangle are
concurrent at a point
(called the centroid)
that is two thirds the
distance from each
C vertex to the
midpoint of the
opposite side.
2
CG  CF
3
2
AG  AD
3
Triangle Altitudes
An altitude of a triangle is a line segment drawn
from any vertex of the triangle to the opposite
side, extended if necessary, and perpendicular to
that side.
B
B
A
E
E
C
BE is an altitude for ΔABC.
A
C
AE is an altitude for ΔABC.
Sides AC and AB are also altitudes. Why?
B
Why?
E
A
F
C
Notice that altitude CF falls outside ΔABC.
Triangle Altitude Theorem
The lines that contain the altitudes of a triangle
are concurrent (at a point called the
orthocenter).
B
A
E
C
Triangle Perpendicular Bisectors Theorem
The perpendicular bisectors of a triangle are concurrent
at a point (called the circumcenter) that is equidistant
from the vertices.
S
QC  RC  SC
X
 Bisector
Bisected  Segments
CX
SX  XQ
CY
SY  YR
CZ
QZ  ZR
Q
C
Z
Y
R
The circle is circumscribed about
the triangle.
Triangle Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent
at a point (called the incenter) that is equidistant from
the sides.
T
XI  YI  ZI
X
I
Y
Bisector
TI
UI
VI
U
Z
V
The circle is inscribed in the
triangle.
Application
M is the centroid of
triangle WOR.
WM=16. Find WX.
W
Y
Z
WX=24
M
R
O
X
Application
U
X
In triangle TUV, Y
is the centroid.
YW=9. Find TY and
TW.
TY=18
W
TW=27
Y
V
T
Z
Application
K
Is KX a median,
altitude, neither, or
both?
both
L
X
M
Application
Find the center of the circle you
can circumscribe about the
triangle with vertices:
A (-4, 5); B (-2, 5); C (-2, -2)
Hint: sketch triangle;
then think about the
perpendicular
bisectors passing
through the midpoints
of the sides!
(-3, 1.5)
Application
Find the center of the circle you
can circumscribe about the
triangle with vertices:
X (1, 1); Y (1, 7); Z (5, 1)
(3, 4)
Try these constructions:
1: Circumscribe a
circle about a triangle
S
•Draw a large triangle.
•Construct the perpendicular
bisectors of any two sides.
The point they meet is the
circumcenter.
•The radius is from the
circumcenter to one of the
vertices. Draw a circle using
this radius and it should pass
through all three vertices.
X
Q
C
Z
Y
R
Try these constructions:
2: Construct a circle inside
a triangle
T
•Draw a large triangle.
•Construct the angle bisectors for
two of the angles. The point they
intersect is called the incenter.
•Drop a perpendicular from the
U
incenter to one of the sides. This is
your radius.
•Draw a circle using this radius and
it should touch each side of the
triangle.
X
I
Z
Y
V