1. Centers of Triangles or Points of Concurrency
Prepared for Ms. Pullo’s
Geometry Classes

2. Medians
Median
vertex to midpoint

3. Example 1 M D P C N What is NC if NP = 18? MC bisects NP…so 18/2 9
If DP = 7.5, find MP. 15 =

4. Three – one from each vertex
How many medians does a triangle have?
Three – one from each vertex

5. They meet in a single point.
The medians of a triangle are concurrent.
The intersection of the medians is called the CENTRIOD.
They meet in a single point.

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. 2x x

7. In ABC, AN, BP, and CM are medians.
Example 2
In ABC, AN, BP, and CM are medians.
If EM = 3, find EC. C
EC = 2(3) N P E
EC = 6 B M A

8. In ABC, AN, BP, and CM are medians.
Example 3
In ABC, AN, BP, and CM are medians.
If EN = 12, find AN. C
AE = 2(12)=24
AN = AE + EN N P
AN = E B
AN = 36 M A

9. In ABC, AN, BP, and CM are medians.
Example 4
In ABC, AN, BP, and CM are medians.
If EM = 3x + 4 and CE = 8x, what is x? A B M P E C N
x = 4

10. In ABC, AN, BP, and CM are medians.
Example 5
In ABC, AN, BP, and CM are medians.
If CM = 24 what is CE? A B M P E C N
CE = 2/3CM
CE = 2/3(24)
CE = 16

11. vertex to side cutting angle in half
Angle Bisector
Angle Bisector
vertex to side cutting angle in half

12. Example 1 W X 1 2 Z Y

13. Example 2 F I G
5(x – 1) = 4x + 1
5x – 5 = 4x + 1
x = 6 H

14. three concurrent Incenter
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 the __________.
Incenter

16. Triangle ADL is a right triangle, so use Pythagorean thm
Example 4
The angle bisectors of triangle ABC meet at point L.
What segments are congruent?
Find AL and FL.
LF, DL, EL
Triangle ADL is a right triangle, so use Pythagorean thm
AL2 =
AL2 = 100
AL = 10 F D E L B C A 8
FL = 6 6

17. vertex to opposite side and perpendicular
Altitude
Altitude
vertex to opposite side and perpendicular

18. The altitude is the “true height” of the triangle.
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of the triangle.
YES NO
YES

19. Three 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
midpoint and perpendicular
(MAY not come from vertex)

21. Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.NO NO
YES

22. Example 2: Find x
3x + 4
5x - 10
x = 7

23. Three 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.
PA = PB = PC

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 to find DP. B
DP = 102
DP = 100
DP2 = 64
DP = 8 6 10 D P A C

26. Tell if the red segment is an altitude, perpendicular bisector, both, or neither?
PER. BISECTOR
BOTH

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
Study!!!