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Warm-Up

Warm-Up. Take out homework p.301 (4-20) even Check your answers with a partner. Medians. Picture:. Both sides are congruent. Median. vertex to midpoint. How many medians can a triangle have?. Median. vertex to midpoint. Midpoint-.

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Warm-Up

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

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

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

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

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

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

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

  8. Medians Picture: Both sides are congruent Median vertex to midpoint

  9. How many medians can a triangle have? Median vertex to midpoint

  10. 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! EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

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

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

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

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

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

  17. Theorem 6-1 Distance from vertex to centroidis twicethe distance fromcentroid to midpoint. Draw Picture 2x x

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

  19. C How much is CX? D CX = 2(XF) E X CX = 2(13) 13 B A F CX = 26

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

  21. C How much is XD? D AX = 2(XD) E X 18 18 = 2(XD) B A F 9 = XD

  22. Ex: 1 In ABC, AN, BP, and CM are medians. If EM = 3, find EC. C N EC = 2(3) P E EC = 6 B M A EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  23. Ex: 2 In ABC, AN, BP, and CM are medians. C If EN = 12, find AN. AE = 2(12)=24 N P E AN = AE + EN B AN = 24 + 12 M A AN = 36 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  24. Altitude Picture: Altitude vertex to opposite side and perpendicular

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

  26. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle. YES NO YES

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

  28. Tell whether each red segment is an perpendicular bisector of the triangle. NO NO YES

  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? 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 Incenter

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

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