2. Warm Up 1.Give the restrictions on the third side of the triangle if the first two sides are 18 and 30. 2. Find x and y for both figures. 55°70° x y 100° 2x + 10 4y 80 1

3. CAAG Unit 7B UNIT QUESTION: What are the properties and characteristics of polygons? Standard: MM1G1.e. Today’s Question: What is the centroid of a triangle, and what is special about it? Standard: MM1G.3.e. 2

4. Special Segments in Triangles 3

5. MedianMedian 4

6. Altitude Altitude 5

7. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle. 6

8. Perpendicular Bisector 7

9. Tell whether each red segment is an perpendicular bisector of the triangle. 8

10. Angle Bisector 9

11. Drill & Practice Indicate which special triangle segment the red line is based on the picture and markings 10

12. 11

13. 12

14. 13

15. 14

16. 15

17. 20  16

18. start at a vertex and bisect the opposite side. Median 17

19. The intersection of the medians is called the CENTROID. How many medians does a triangle have? 18

20. 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. 19

21. A B F X E C D 20

22. A B F X E C D 21

23. 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 22

24. 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 23

25. Altitude 24

26. The intersection of the altitudes is called the ORTHOCENTER. How many altitudes does a triangle have? 25

27. 26

28. The intersection of the perpendicular bisectors is called the CIRCUMCENTER. How many perpendicular bisectors does a triangle have? 27

29. What is special about the CIRCUMCENTER? Equidistant to the vertices of the triangle. 28

30. Example 1: Point G is the circumcenter of the triangle. Find GB. B A C G E D F 2 5 7 7 29

31. Angle Bisector 30

32. The intersection of the angle bisectors is called the INCENTER. How many angle bisectors does a triangle have? 31

33. What is special about the INCENTER? Equidistant to sides of the triangle 32

34. Example 1: Point N is the incenter of the triangle. Find the length of segment ON. 18 30 18 33

35. Memorize these! MC AO ABI PBCC Medians/Centroid Altitudes/Orthocenter Angle Bisectors/Incenter Perpendicular Bisectors/Circumcenter 34

36. Will this work? MC AO ABI PBCC My Cat Ate Our Apples But I Prefer Blue Cheese Crumbles 35

37. Come up with your own! 36