Concurrent Lines, Medians & Altitudes

1. Concurrent Lines, Medians & Altitudes Geometry Honors

2. Vocabulary Concurrent Lines – when three or more lines intersect in one point. Point of concurrency– the point at which 3 or more lines intersect.

3. Geogebra Demonstration of Perpendicular Bisectors

4. Vocabulary Circumcenter of the triangle– the point of concurrency of the perpendicular bisectors. Circumcenter

5. Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

6. Since the vertices of the triangle are equidistant from the circumcenter, we can draw a circle around the triangle or circumscribe the triangle. The center of the circle is the circumcenter of the triangle.

7. Geogebra Demonstration of Angle Bisectors

8. Vocabulary Incenter of the triangle– the point of concurrency of the angle bisectors. Incenter

9. Theorem The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

10. We can now inscribe a circle in the triangle since the incenter is equidistant from the sides. The center of the circle is the incenter of the triangle.

11. Name the point of concurrency of the angle bisectors. Q is the incenter of this triangle, because it is where the two angle bisectors intersect.

12. Name the point of concurrency of the angle bisectors. Z is the incenter of this triangle, because it is where the two angle bisectors intersect.

13. AB and DF are angle bisectors. Therefore, F is the incenter. So FE must equal FD. A E F B C D Find x? x= 2

14. CE and AF are angle bisectors. Therefore, D is the incenter. So DE must equal DF. B E A F D C Find x? x = 4

15. The towns of Adamsville, Brooksville and Cartersville want to build a library that is equidistant from the three towns. The three towns are located in a triangular pattern. Where should they build the library…at the circumcenter or the incenter? circumcenter

16. Vocabulary Median of a triangle– a segment whose endpoints are a vertex and the midpoint of the opposite side. Every triangle has three medians.

17. Vocabulary Centroidof the triangle– the point of concurrency of the medians of a triangle. Centroid

18. FYI The centroid of the triangle is the center of gravity of the triangle. It is the point where a triangular shape will balance.

19. Theorem The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. A AG = 2/3 AF E D BG = 2/3 BE G CG = 2/3 CD C B F

20. If A is the centroid of XYZ, and DZ = 12, find AD and AZ. AZ = 2/3 DZ AD + AZ = DZ AZ = 2/3 (12) AD + 8 = 12 AZ = 8 AD = 4

21. If A is the centroid of XYZ, and AB= 6, find AY and BY. AB = 1/3 BY AY = 2/3 BY 6 = 1/3 (BY) AY = 2/3 (18) AY =12 BY = 18

22. Vocabulary Altitude of a triangle– the perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes.

23. FYI The altitude of a triangle may be inside the triangle.

24. FYI The altitude of a triangle may be the side of a triangle.

25. FYI The altitude of a triangle may be outside of a triangle.

26. Vocabulary Orthocenter of the triangle– the point of concurrency of the altitudes of a triangle. Orthocenter

27. Theorem The lines that contain the altitudes are concurrent.

28. Name an altitude in STU. AU SB CB UD Name a median in STU. Name a median in SBU. Name an altitude in CBU.

29. Find the orthocenter of ABD. H

30. Which triangle has the centroid at the same point as the orthocenter? GHI

31. Tell which line contains the circumcenter of ABC. k Tell which line contains the incenter of ABC. n

32. Tell which line contains the orthocenter of ABC. m Tell which line contains the centroid of ABC. l

33. Homework