1. Lesson 5-1 Bisectors, Medians, and Altitudes

2. Ohio Content Standards:

3. Formally define geometric figures.

4. Ohio Content Standards: Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);

5. Perpendicular Bisector

6. A line, segment, or ray that passes through the midpoint of the side of a triangle and is perpendicular to that side.

7. Theorem 5.1

8. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

9. Example CD A B

10. Theorem 5.2

11. Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

12. Example CD A B

13. Concurrent Lines

14. When three or more lines intersect at a common point.

15. Point of Concurrency

16. The point of intersection where three or more lines meet.

17. Circumcenter

18. The point of concurrency of the perpendicular bisectors of a triangle.

19. Theorem 5.3 Circumcenter Theorem

20. The circumcenter of a triangle is equidistant from the vertices of the triangle.

21. Example C A B circumcenter K

22. Theorem 5.4

23. Any point on the angle bisector is equidistant from the sides of the angle. A C B

24. Theorem 5.5

25. Any point equidistant from the sides of an angle lies on the angle bisector. A B C

26. Incenter

27. The point of concurrency of the angle bisectors.

28. Theorem 5.6 Incenter Theorem

29. The incenter of a triangle is equidistant from each side of the triangle. C A Bincenter K P Q R

30. Theorem 5.6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. C A Bincenter K P Q R If K is the incenter of ABC, then KP = KQ = KR.

31. Median

32. A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

33. Centroid

34. The point of concurrency for the medians of a triangle.

35. Theorem 5.7 Centroid Theorem

36. The centroid of a triangle is located two-thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

37. Example C A B DL E F centroid

38. Altitude

39. A segment from a vertex in a triangle to the line containing the opposite side and perpendicular to the line containing that side.

40. Orthocenter

41. The intersection point of the altitudes of a triangle.

42. Example C A B D L E F orthocenter

43. Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c. Y W U X V Z 7.4 5c 8.7 15.2 2a2a 3b + 2

44. The vertices of  QRS are Q(4, 6), R(7, 2), and S(1, 2). Find the coordinates of the orthocenter of  QRS.

45. Assignment: Pgs. 243-245 13-20 all