1. Centers of Triangles or Points of Concurrency

2. homework
Point of Concurrency Worksheet page 2

3. Point of concurrency
A point of concurrency is where three or more lines intersect in one place.
Construction: the three angle bisectors, medians, perpendicular bisectors, and altitudes are concurrent in every triangle.

5. Median Medians vertex to midpoint
Medians
Median
vertex to midpoint

6. 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 =

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

8. 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.

9. 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

10. 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

11. 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

12. 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

13. 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

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

15. Example 1 W X 1 2 Z Y

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

17. 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

18. Properties of the Incenter
The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides.
Always inside the triangle
CLICK HERE

19. The incenter is the same distance from the sides of the triangle.
Point P is called the __________.
Incenter

20. 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

21. Perpendicular Bisector
midpoint and perpendicular
(don't care about no vertex)

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

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

24. 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.

25. Circumcenter Where is the circumcenter located on An acute triangle
An obtuse triangle
A right triangle
Inside the triangle
Outside the triangle
On the triangle

26. The Circumcenter is equidistant from the vertices of the triangle.
PA = PB = PC

27. 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

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

29. 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

30. Three How many altitudes does a triangle have?
The altitudes of a triangle are concurrent.
The intersection of the altitudes is called the ORTHOCENTER.

31. Orthocenter
The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside.
If the triangle is right it will be on the vertex of the right angle
CLICK HERE

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

33. Aim: The four centers of a triangle
Do Now: Match the description with the correct center of a triangle.
The three altitudes meet at the
Circumcenter
The perpendicular bisectors of the three sides meet at the
Centroid
The three medians meet at the
Incenter
The angle bisectors meet at the
Orthocenter