# Circumcentre, Incentre, Orthocentre and Centroids

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This ppt includes the following topics:-
Medians
Centroid
Angle Bisector
Incentre
Altitude
Orthocentre
Perpendicular Bisector and many more.
1. Centers of Triangles or
Points of Concurrency
Prepared for Ms. Pullo’s
Geometry Classes
2.
3. Example 1
In MNP, MC and ND are medians.
M D P
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
4. How many
medians does a
triangle have?
5. The medians of
a triangle are
concurrent.
The intersection of
the medians is
called the
CENTRIOD.
6. 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.
7. Example 2
In ABC, AN, BP, and CM are medians.
If EM = 3, find C
EC = N
P
2(3) E
EC = 6 B
M
A
8. Example 3
In ABC, AN, BP, and CM are medians.
If EN = 12, find
AE = C
AN = AE + EN
2(12)=24 N
P E
AN = 24 +
B
12
AN = A M
9. Example 4
In ABC, AN, BP, and CM are medians.
If EM = 3x + 4
and CE = 8x, C
what is x?
N
P E
x=4 M
B
A
10. Example 5
In ABC, AN, BP, and CM are medians.
If CM = 24
what is CE? C
CE = 2/3CM N
CE = P E
2/3(24) B
CE = 16 M
A
11. Angle Bisector
12. Example 1
In WYZ, ZX bisects WZY . If m1 = 55,
find mWZY .
W mWZY 55  55
mWZY 110
X
1
2
Z Y
13. Example 2
In FHI, IG is an angle bisector. Find mHIG.
F
5( x  1) I
G (4 x  1)
5(x – 1) = 4x + 1
H 5x – 5 = 4x + 1
x=6
14. 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
15. The incenter is the same distance from the
sides of the triangle.
Point P is called
B
the __________.
Incenter
D
F P
A E C
16. Example 4
The angle bisectors of triangle ABC meet at point L.
• What segments are congruent? LF, DL, EL
• Find AL and FL.
Triangle ADL is a
A right triangle, so use
FL = 6 Pythagorean thm
8
D AL2 = 82 + 62
F AL2 = 100
L AL = 10
6
C E B
17.
18. Tell whether each red segment is an altitude of the
The altitude is the
“true height” of
the triangle.
19. How many altitudes
does a triangle have?
The altitudes of
a triangle are
concurrent.
The intersection of the
altitudes is called the
ORTHOCENTER.
20. Perpendicular Bisector
21. Example 1: Tell whether each red segment is
a perpendicular bisector of the triangle.
22. Example 2: Find x
3x + 4 5x - 10
23. 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.
24. The Circumcenter is
equidistant from the vertices
of the triangle.
B
PA = PB = PC
P
A C
25. Example 3: The perpendicular bisectors of
triangle ABC meet at point P.
• Find DA. DA = 6
• Find BA. BA = 12
• Find PC. PC = 10
• Use the Pythagorean Theorem B
to find DP.
DP2 + 62 = 102 6
D 10
DP + 36 = 100
2
DP2 = 64 P
A C
DP = 8
26. Tell if the red segment is an altitude,
perpendicular bisector, both, or neither?
NEITHER
ALTITUDE
PER.
BOTH BISECTOR
27. IN A NUT SHELL
Median – Centroid
Angle Bisector – Incenter
Altitude – Orthocenter
Perpendicular Bisector - Circumcenter
Angle Bisector: The Incentor is equidistance to
the sides
Perpendicular Bisector – the Circumcenter is
equidistance to the
vertex
28. The End