2. Points of Concurrency MM1G3e Students will be able to find and use points of concurrency in triangles.

3. Median of a Triangle A segment from one vertex of the triangle to the midpoint of the opposite side.

4. The intersection of the medians is called the CENTROID. How many medians does a triangle have?

5. Theorem 5.8 The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

6. A B F X E C D

7. A B F X E C D

8. In  ABC, AN, BP, and CM are medians. A B M P E C N If EM = 3, find EC. EC = 2(3) Ex: 1 EC = 6

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

10. In  ABC, AN, BP, and CM are medians. A B M P E C N If CM = 3x + 6, and CE = x + 12, what is x? CM = CE + EM Ex: 3 x = 8 3x + 6 = (x + 12) +.5(x + 12) 3x + 6 = x + 12 +.5x + 6 3x + 6 = 1.5x + 18 1.5x = 12

11. Altitude

12. The intersection of the altitudes is called the ORTHOCENTER. How many altitudes does a triangle have?

13. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle.

15. A E C B D G 1. If CD = 3.25, what is BC? 2. Find AG if DG = 10. 3. If CG = 7, find CE? Altitude, perpendicular bisector, both, or neither? 6.5 20 10.5 ALTITUDE NEITHER BOTH PER. BISECTOR

16. Homework Answers page 280 1-6, 10-14 1.8 2.16 3.5 4.15 5.12 6.6 10. Yes, yes, yes 11. No, no, no 12. No, yes, no 13. 12, 78 o 14. 6.5, 15

19. The intersection of the perpendicular bisector is called the CIRCUMCENTER. How many perpendicular bisectors does a triangle have?

20. What is special about the CIRCUMCENTER? Equidistant to the vertices of the triangle.

21. Example 1: Point G is the circumcenter of the triangle. Find GB. B A C G E D F 2 5 7 GB=7

22. Example 2: Point G is the circumcenter of the triangle. Find CG. B A C G E D F 6 8 CG=10

23. Angle Bisector

24. The intersection of the angle bisectors is called the INCENTER. How many angle bisectors does a triangle have?

25. What is special about the INCENTER? Equidistant to sides of the triangle

26. Example 1: Point N is the incenter of the triangle. Find the length of segment ON. ON=18 30 18

27. Example 2: Point N is the incenter of the triangle. Find the length of segment NP. NP=15

28. p. 266 #13-18 p. 275 #14-17