1. 6.2 & 6.3 Triangle Concurrency Points
Geometry
6.2 & 6.3 Triangle Concurrency Points

2. 6.2 and 6.3 Points of Concurrency
Essential Question
What are points of concurrency?
November 28, 2018
6.2 and 6.3 Points of Concurrency

3. 6.2 and 6.3 Points of Concurrency
Concurrent Lines
Three or more lines intersecting at the same point are concurrent.
The point where they intersect is the point of concurrency.
November 28, 2018
6.2 and 6.3 Points of Concurrency

4. Perpendicular Bisector
The perpendicular bisector of a line segment is the line that is perpendicular to the segment at its midpoint. A B R S
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6.2 and 6.3 Points of Concurrency

5. 6.2 and 6.3 Points of Concurrency
Angle Bisector
The angle bisector is a ray that divides an angle into two congruent adjacent angles. A D C B
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6.2 and 6.3 Points of Concurrency

6. 6.2 and 6.3 Points of Concurrency
Altitude
An altitude of a triangle is a segment drawn from a vertex perpendicular to the opposite side (or to the line containing the opposite side).
A triangle has three altitudes.
November 28, 2018
6.2 and 6.3 Points of Concurrency

7. 6.2 and 6.3 Points of Concurrency
Median
A median of a triangle is the segment drawn from a vertex to the midpoint of the opposite side.
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6.2 and 6.3 Points of Concurrency

8. 6.2 and 6.3 Points of Concurrency
Four points of concurrency:
Circumcenter
Incenter
Centroid
Orthocenter
Each is made using different types of lines and has its own properties.
November 28, 2018
6.2 and 6.3 Points of Concurrency

9. Concurrent Point: Circumcenter
Created when using the perpendicular bisectors of each side of a triangle.
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6.2 and 6.3 Points of Concurrency

10. Concurrent Point: Circumcenter
When all three perpendicular bisectors are drawn, the point of concurrency created is called the circumcenter.
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6.2 and 6.3 Points of Concurrency

11. Circumcenter Property
The circumcenter is equidistant from each vertex of the triangle.
This is called a circumcircle.
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6.2 and 6.3 Points of Concurrency

12. Concurrent Point: Incenter
Created when using the angle bisectors of each vertex of a triangle.
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6.2 and 6.3 Points of Concurrency

13. Concurrent Point: Incenter
When all three angle bisectors are drawn, the point of concurrency created is called the incenter.
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6.2 and 6.3 Points of Concurrency

14. 6.2 and 6.3 Points of Concurrency
Incenter Property
The incenter is equidistant from the sides of a triangle.
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6.2 and 6.3 Points of Concurrency

15. 6.2 and 6.3 Points of Concurrency
Incenter Property
The incenter is equidistant from the sides of a triangle.
This is called an incircle.
November 28, 2018
6.2 and 6.3 Points of Concurrency

16. Concurrent Point: Centroid
Created when using the medians of a triangle.
November 28, 2018
6.2 and 6.3 Points of Concurrency

17. Concurrent Point: Centroid
When all three medians are drawn, the point of concurrency created is called the centroid.
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6.2 and 6.3 Points of Concurrency

18. 6.2 and 6.3 Points of Concurrency
Centroid Property
The centroid of a triangle is two thirds of the distance from each vertex to the midpoint of the opposite side. A B C
November 28, 2018
6.2 and 6.3 Points of Concurrency

19. Concurrency Point: Orthocenter
When all three altitudes are drawn, the point of concurrency created is called the orthocenter.
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6.2 and 6.3 Points of Concurrency

20. 6.2 and 6.3 Points of Concurrency
Orthocenter Property
None!
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6.2 and 6.3 Points of Concurrency

21. 6.2 and 6.3 Points of Concurrency
Centroid Property
The centroid of a triangle is also known as the center of balance.
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6.2 and 6.3 Points of Concurrency

22. 6.2 and 6.3 Points of Concurrency
Centroid Property
The centroid of a triangle is two thirds of the distance from each vertex to the midpoint of the opposite side. A B C
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6.2 and 6.3 Points of Concurrency

23. 6.2 and 6.3 Points of Concurrency
Examples
Turn to your notes to practice a few examples.
C is the centroid of the triangle. C S R
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6.2 and 6.3 Points of Concurrency

24. 6.2 and 6.3 Points of Concurrency
Centroid Property R x C S
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6.2 and 6.3 Points of Concurrency

25. 6.2 and 6.3 Points of Concurrency
Example 1 C S R
RS = 9
RC = ?
CS = ? 6 3
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6.2 and 6.3 Points of Concurrency

26. 6.2 and 6.3 Points of Concurrency
Example 2 C S R
RS = 15
RC = ?
CS = ? 10 5
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6.2 and 6.3 Points of Concurrency

27. 6.2 and 6.3 Points of Concurrency
Example 3 C S R
RS = ?
RC = 8
CS = ? 8 12 4
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6.2 and 6.3 Points of Concurrency

28. 6.2 and 6.3 Points of Concurrency
Example 4 C S R
RS = ?
RC = 30
CS = ? 30 45 15
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6.2 and 6.3 Points of Concurrency

29. 6.2 and 6.3 Points of Concurrency
Example 5 C S R
RS = ?
RC = ?
CS = 7 14 21 7
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6.2 and 6.3 Points of Concurrency

30. 6.2 and 6.3 Points of Concurrency
Example 6 C S R
RS = ?
RC = ?
CS = 4.2
12.6
8.4
4.2
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6.2 and 6.3 Points of Concurrency

31. 6.2 and 6.3 Points of Concurrency
Fast answers!
The altitudes are concurrent at the ?
Orthocenter
The medians are concurrent at the ?
Centroid
The perpendicular bisectors are concurrent at the ?
Circumcenter
The angle bisectors are concurrent at the ?
Incenter
November 28, 2018
6.2 and 6.3 Points of Concurrency

32. 6.2 and 6.3 Points of Concurrency
Fast answers!
Which point is equidistant from the sides of a triangle?
Incenter created by ∠ Bisectors
Which point is the center of balance?
Centroid created by Medians
Which point is equidistant from the vertices?
Circumcenter created by ⊥ Bisectors
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6.2 and 6.3 Points of Concurrency

33. 6.2 and 6.3 Points of Concurrency
Fast answers!
What point is needed to draw a circumcircle?
Circumcenter created by ⊥ Bisectors
What point is needed to draw an incircle?
Incenter created by ∠ Bisectors
What point is needed to find the center of balance?
Centroid created by the Medians
November 28, 2018
6.2 and 6.3 Points of Concurrency

34. Centroid in the Coordinate Plane
The coordinates of the centroid can be found using the formula:
𝑥 1 + 𝑥 2 + 𝑥 3 3 , 𝑦 1 + 𝑦 2 + 𝑦 3 3
Which is the average of the
three vertices.
November 28, 2018
Geometry 5.1 Perpendiculars and Bisectors

35. Centroid in the Coordinate Plane
Find the centroid of the triangle with vertices (2, 5), (7, 8), & (9, 2). ,
18 3 , 15 3
Centroid: 6, 5
November 28, 2018
Geometry 5.1 Perpendiculars and Bisectors

36. Geometry 5.1 Perpendiculars and Bisectors
Example 1 
 ∆MNP has vertices M (–4, –3), N (6, –1), 
and P (–11, 10).   Find the centroid.
−4+6−11 3 , −3−1+10 3
−9 3 , 6 3
Centroid: −3, 2
November 28, 2018
Geometry 5.1 Perpendiculars and Bisectors

37. Geometry 5.1 Perpendiculars and Bisectors
Example 2 
A sculptor is shaping a triangular piece of iron, with coordinates (6, 6), (10, 7), and (8, 2), that will balance on the point of a cone. At what coordinates will the triangular region balance?  ,
24 3 , 15 3
Centroid: 8, 5
November 28, 2018
Geometry 5.1 Perpendiculars and Bisectors

38. Now add these to the back of your homework.
Find the coordinates of the centroid for each triangle.
𝐴 2, 3 , 𝐵 8, 1 , 𝐶 5,7
2. 𝐹 1, 5 , 𝐺 −2, 7 , 𝐻 −6, 3
3. 𝑆 5, 5 , 𝑇 11, 3 , 𝑈(−1, 1)
4. 𝑋 1, 4 , 𝑌 7, 2 , 𝑍(2, 3)
November 28, 2018
Geometry 5.1 Perpendiculars and Bisectors

39. Concurrent Point: Orthocenter
Turn back to the graphic organizer. Created when using the altitudes of a triangle. In the example box, draw one of the altitudes of the triangle.
November 28, 2018
6.2 and 6.3 Points of Concurrency