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Triangles: Points of Concurrency

Triangles: Points of Concurrency. MM1G3 e. Investigate Points of Concurrency. http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html. Circumcenter. Perpendicular Bisectors and Circumcenters Examples.

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Triangles: Points of Concurrency

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  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

  18. Medians and Centroids Examples

  19. 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.

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

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

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

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

  24. 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

  25. 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

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

  27. 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.

  28. Centroids • Investigate • Construction

  29. Try These: D

  30. Try These: C

  31. Try These: A

  32. Try These: C

  33. Angle Bisectors and Incenters Examples

  34. 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.

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

  36. 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

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

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

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

  40. 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

  41. 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.

  42. Incenter • Investigate • Construction

  43. Try These: B

  44. Try These: C

  45. Try These: C

  46. Try These: K C

  47. Altitudes and Orthocenters Examples

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