Triangles: Points of Concurrency

1. Triangles: Points of Concurrency MM1G3e

2. Investigate Points of Concurrency • http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html

3. Circumcenter

4. Perpendicular Bisectors and Circumcenters Examples

5. A perpendicular bisector of a triangle is a line or line segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisectorof the side. A B C D

6. Example 1: A F E C B D

7. Example 2: M Q N P

8. Since a triangle has three sides, it will have three perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter. D G is the circumcenter of ∆DEF. Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices. G E F

9. The circumcenter will be located inside an acute triangle (fig.1),outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters. fig. 1 fig. 3 fig. 2

10. Example 3: A company plans to build a new distribution center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?

11. The circumcenter is equidistant to all three vertices of a triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same. C

12. In Summary • The circumcenter is the point where the three perpendicular bisectors of a triangle intersect. • The circumcenter can be inside, outside, or on the triangle. • The circumcenter is equidistant from the vertices of the triangle

13. Circumcenter • Exploration • Construction

14. Try These: C

15. Try These: D

16. Try These: A

17. Try These B

19. Medians and Centroids Examples

20. A median of a triangle is a line segment that contains the vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.

21. Since a triangle has three sides, it will have three medians. These medians will meet at a common point – the centroid.

22. The centroid is always located inside the triangle. Acute triangle

23. Q The distance from any vertex to the centroid is 2/3 the length of the median. E F G S R D

24. Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3. Find QD and SF. Q E F G S R D

25. Example 3: G is the centroid of triangle DEF. FG = 15, ES = 21, QG = 5 Determine FR, EG and GD E Q 5 G 15 F R 21 S D

26. Notice that the distance from any vertex to the centroid is 2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median. So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ M V N Q U T P

27. The centroid is also known as the balancing point (center of gravity) of a triangle.

28. In Summary • A median is a line segment from the a vertex of a triangle to the midpoint of the opposite side. • The distance from the vertex to the centroid is 2/3 the length of the median. • The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid. • Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.

29. Centroids • Investigate • Construction

30. Try These: D

31. Try These: C

32. Try These: A

33. Try These: C

35. Angle Bisectors and Incenters Examples

36. An angle bisector of a triangle is a segment that shares a common endpoint with an angle and divides the angle into two equal parts.

37. Example 1: Determine any angle bisectors oftriangle ABC. A G F E C B D

38. Since a triangle has three angles, it will have three angle bisectors. These angle bisectors will meet at a common point – the incenter. X M Z Y

39. The incenter is always located inside the triangle. incenter Acute triangle Obtuse triangle Right triangle

40. The incenter is equidistant to the sides of the triangle. x x

41. Example 2: L is the incenter of triangle ABC. Which segments are congruent? A D B L E F C

42. Example 3: Given P is the incenter of triangle RST. PN = 10 and MT = 12, find PM and PT. 12 10 Not drawn to scale

43. In Summary • The incenter is the point of intersection of the three angle bisectors of a triangle. • The incenter is equidistant to all three sides of the triangle.

44. Incenter • Investigate • Construction

45. Try These: B

46. Try These: C

47. Try These: C

48. Try These: K C

50. Altitudes and Orthocenters Examples