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Warm up

Warm up. Will the following side lengths make a triangle? 2, 4, 5 4, 3, 1 2 . Find the range of the third side of the triangle: 1, 3, x 4, 8, x. Warm up Part 2. In Triangle DOG, side DO = 25, side OG = 15 and side DG = 30. List the angles in descending order:

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Warm up

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  1. Warm up • Will the following side lengths make a triangle? • 2, 4, 5 • 4, 3, 1 • 2. Find the range of the third side of the triangle: • 1, 3, x • 4, 8, x

  2. Warm up Part 2 • In Triangle DOG, side DO = 25, side OG = 15 and side DG = 30. List the angles in descending order: • 2. In Triangle CAT, <C is 60o , <T is 25o .List the sides in ascending order:

  3. POP QUIZ! You have 15 minutes to complete this.

  4. Angle Bisector Angle Bisector cuts the angle into 2 equal parts

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

  6. Incenter Point of Concurrency of the Angle Bisectors Always INSIDE the triangle! Equidistant from the SIDES of a triangle

  7. EX:1 A 41 1 2 D B C

  8. EX:2 K 45 L 1 2 M N

  9. EX:3 W X 1 2 110 Z Y

  10. EX:4 F 25 4(6)+1=24+1= I G H

  11. Perpendicular Bisector Perpendicular Bisector midpoint and perpendicular

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

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

  14. Circumcenter Point of Concurrency of the Perpendicular Bisectors Can be inside, outside, or on the triangle. Equidistant from the VERTICES of a triangle

  15. Medians Median vertex to midpoint

  16. M D P What is NC if NP = 18? C MC bisects NP…so 18/2 9 If DP = 7.5, find MP. N 7.5 + 7.5 = 15

  17. A E B D C If CD = 2x + 5, BD = 4x – 1, SOLVE FOR X. BD = CD 4x - 1 = 2x + 5 2x = 6 x = 3

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

  19. When 3 or more lines (or rays or segments) intersect in the same point, they are called concurrent lines (or rays or segments). The point of intersection of lines is called the point of concurrency.

  20. Centroid Point of Concurrency of the Medians Always INSIDE the triangle.

  21. Theorem The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. 2x x

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

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

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

  25. 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

  26. Answer Me: E A B G D C 1. If CD = 3.25, what is BC? 6.5 2. Find AG if DG = 10. 20 3. If CG = 7, find CE? 10.5

  27. Altitude Altitude vertex to opposite side and perpendicular

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

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

  30. Orthocenter Point of Concurrency of the Altitudes Can be inside, outside, or on the triangle.

  31. Now we must come up with a silly way to remember the points of concurrency. It's up to you!

  32. MEDIAN = CENTROID MC What's something or someone that has these initials?

  33. ALTITUDE = ORTHOCENTER AO

  34. PERPENDICULAR BISECTORS = CIRCUMCENTER PBCC(V)

  35. ANGLE BISECTORS = INCENTER ABI(S)

  36. Some Special Cases... Equilateral Triangle Right Triangle

  37. Euler Line The orthocenter, circumcenter, and the centroid are COLLINEAR in EVERY triangle!

  38. TOTD E A 1. If CD = 3.25, what is BC? B 6.5 G D C Altitude, perpendicular bisector, both, or neither? PER. BISECTOR NEITHER BOTH ALTITUDE

  39. Homework Pg 335 # 3 - 10 Learn your vocabulary!!!

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