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Points of Concurrency

Points of Concurrency. MM1G3e Students will be able to find and use points of concurrency in triangles. Median of a Triangle. A segment from one vertex of the triangle to the midpoint of the opposite side. How many medians does a triangle have?.

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Points of Concurrency

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

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

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

  4. 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. 2x x

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

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

  7. 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

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

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

  10. Altitude Altitude vertex to opposite side and perpendicular

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

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

  13. HWK: Textbook p. 280 1 - 6, 10-14

  14. Warm-up #37 E A B 1. If CD = 3.25, what is BC? 6.5 G 2. Find AG if DG = 10. D 20 C 3. If CG = 7, find CE? 10.5 Altitude, perpendicular bisector, both, or neither? PER. BISECTOR NEITHER BOTH ALTITUDE

  15. 8 16 5 15 12 6 10. Yes, yes, yes 11. No, no, no 12. No, yes, no 13. 12, 78o 14. 6.5, 15 Homework Answerspage 280 1-6, 10-14

  16. 5-2 Perpendicular Bisector perpendicular to a side at its midpoint

  17. 5-2 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

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

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

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

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

  22. 5-3 Angle Bisector Angle Bisector cuts the angle into 2 equal parts

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

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

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

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

  27. Homework p. 266 #13-18 p. 275 #14-17 Learn your vocabulary!!!

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