Transcript Centers of Triangles or Points of Concurrency
Centers of Triangles or Points of Concurrency
Median
Example 1
In
MNP
, MC and ND are medians.
M D P N C What is NC if NP = 18?
MC bisects NP…so 18/2 9 If DP = 7.5, find MP.
7.5 + 7.5 = 15
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.
Example 2
In ABC, AN, BP, and CM are medians.
If
EM
= 3, find
EC
.
EC = 2(3)
EC = 6
P C E M A N B
Example 3
In ABC, AN, BP, and CM are medians.
If
EN
= 12, find
AN
.
AE = 2(12)=24 AN = AE + EN AN = 24 + 12
P
AN = 36
A C E M N B
Example 4
In ABC, AN, BP, and CM are medians.
If EM CE = 3x + 4 and = 8x, what is x?
C N P E
x = 4
M B A
Example 5
In ABC, AN, BP, and CM are medians.
If CM CE?
= 24 what is CE = 2/3CM CE = 2/3(24) CE = 16
P C E N B M A
Angle Bisector
Example 1
In WYZ, ZX bisects
WZY
find m
WZY
.
W X Z 1 2
110
Y
Example 2
In FHI, IG is an angle bisector. Find m
HIG
.
F G H
5 (
x
( 4
x
I
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 The intersection of the angle bisectors is called the Incenter
The incenter is the same distance from the sides of the triangle.
A B F P D E C
Point P is called the Incenter
Example 4 The angle bisectors of triangle ABC meet at point L. • What segments are congruent?
LF, DL, EL • Find AL and FL. FL = 6
A 8
Triangle ADL is a right triangle, so use Pythagorean thm
D
AL 2 = 8 2 + 6 2
F
AL 2 = 100
C 6 L E B
AL = 10
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 PA = PB = PC P A C
Example 3: The perpendicular bisectors of triangle ABC meet at point P.
•
Find DA.
DA = 6 • • •
Find BA.
Find PC.
BA = 12 PC = 10
Use the Pythagorean Theorem to find DP.
B
DP 2 + 6 2 = 10 2
6 10
DP 2 + 36 = 100
D
DP 2 = 64
P
DP = 8
A C
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?
NEITHER ALTITUDE BOTH PER. BISECTOR