Centers of Triangles or Points of
Concurrency
Median
M D P
N
C
In MC and ND are medians.MNP,
What is NC if NP = 18?
9MC bisects NP…so 18/2
If DP = 7.5, find MP.
7.5 + 7.5 = 15
Example 1
How many medians does a triangle have?
The medians of a triangle are concurrent.
The intersection of the medians is
called the CENTRIOD.
Theorem
The length of the segment from the vertex to the
centroid is twice the length of the segment from the centroid to the midpoint.
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
N
If EM = 3, find EC.EC = 2(3)EC = 6
Example 2
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
N
If EN = 12, find AN.AE = 2(12)=24
AN = 36
AN = AE + ENAN = 24 + 12
Example 3
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
N
If EM = 3x + 4 and CE = 8x, what is x?
x = 4
Example 4
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
N
If CM = 24 what is CE?
CE = 2/3CM
CE = 2/3(24)
CE = 16
Example 5
Angle Bisector
X
W
Z Y1
2
Example 1
In WYZ, ZX bisects . If m 1 = 55,
find m .
WZY
WZY
55 55
110
m WZY
m WZY
G
F
H
I
Example 2In FHI, IG is an angle bisector. Find m . HIG
5 1( )x ( )4 1x
5(x – 1) = 4x + 1
5x – 5 = 4x + 1
x = 6
How many angle bisectors does a triangle have?
three
The angle bisectors of a triangle are ____________.concurrent
The intersection of the angle bisectors is called the ________.Incenter
The incenter is the same distance from the sides of the triangle.
B
A C
P
E
D
F
Point P is called the __________.Incenter
Example 4
The angle bisectors of triangle ABC meet at point L. • What segments are congruent?• Find AL and FL.
F
D
E
L
BC
A
8
LF, DL, EL
FL = 6
Triangle ADL is a right triangle, so use
Pythagorean thm
AL2 = 82 + 62
AL2 = 100
AL = 106
Perpendicular Bisector
Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.
Example 2: Find x
3x + 4 5x - 10
How many perpendicular bisectors does a triangle
have?
The perpendicular bisectors of a triangle
are concurrent.
The intersection of the perpendicular bisectors is called
the CIRCUMCENTER.
The Circumcenter is equidistant from the vertices
of the triangle.B
A C
P
PA = PB = PC
6
D
B
A C
P
10
Example 3: The perpendicular bisectors of triangle ABC meet at point P.
• Find DA.• Find BA.• Find PC.• Use the Pythagorean Theorem
to find DP.
DA = 6
BA = 12
PC = 10
DP2 + 62 = 102
DP2 + 36 = 100
DP2 = 64
DP = 8
Altitude
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of
the triangle.
How many altitudes does a triangle have?
The altitudes of a triangle are concurrent.
The intersection of the altitudes is called the
ORTHOCENTER.
Tell if the red segment is an altitude, perpendicular bisector, both, or neither?
ALTITUDE
NEITHER
BOTH
PER. BISECTOR