1. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? Warm-Up Take out homework p.301 (4-20) even Check your answers with a partner

2. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

3. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

4. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

5. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

6. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

7. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

8. Median Picture: Both sides are congruent

9. Median

10. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? Midpoint- If you have a midpoint- then the segments on both sides are CONGRUENT! That is why you will see the “tick marks” For the measure of the entire line- add both sides!

11. MDP N C 1. What is NC if NP = 18? 9 2. If DP = 7.5, find MP. 15

12. ABC E D 1.What is ED if DC = 14? 2.What Is AC is BC is 9? 14 3.If BC = 3, find AC. 6 AD and EB are medians:

13. So if you are given the length of the entire side, how do you find a missing segment? If you are given the length of a segment, how to you find the entire side? If you have a median- what do you know about each side?

14. AE B C D If CD = 2x + 5, BD = 4x – 1, and AE = 5x –2, find BE. BD = CD 4x – 1= 2x + 5 2x = 6 x = 3 AE = BE BE = 5x – 2 BE = 5(3) – 2 BE = 13

15. The intersection of the medians is called the CENTROID. How many medians does a triangle have? Draw the Picture:

16. When three or more lines or segments meet at the same point. Concurrent:

17. Theorem 6-1 Distance from vertex to centroid is twice the distance from centroid to midpoint. Draw Picture

18. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? Vertex to Centroid  LONGER Centroid to Midpoint  shorter x + 2x = whole median

19. A B F X E C D

20. Quick Assessment What is a median? What is a centroid? What does concurrent mean? What is the vertex? What is a midpoint? 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

21. A B F X E C D

22. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? 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

23. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? 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

24. Altitude Picture:

25. 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors? Altitude The altitude is the “true height” of the triangle.

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

27. Perpendicular Bisector Picture: Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector

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

29. Can you have both? Can it be both an altitude and perpendicular bisector?

30. Quick Assessment What is the difference between altitude and perpendicular bisector? What is an altitude? What is a perpendicular bisector? How can the segment be both- altitude and perpendicular bisector? 8/9/2015 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

31. Angle bisector of a triangle : is the bisector of each angle of the triangle. (cuts each angle in half) Incenter of the triangle: where the angle bisectors meet.

32. Angle Bisectors Incent er

33. Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. B F A D P E C PD = PE = PF