1. Chapter 5: Relationships within Triangles 5.3 Concurrent Lines, Medians, and Altitudes

2. Activity I will give you a triangle (acute, obtuse, or right) and a ruler. Fold the triangles to create the angle bisectors of each angle of the triangle. What do you see about the angle bisectors? Make a conjecture about the bisectors of the angles of a triangle.

3. Activity I will give you a second triangle (acute or right). Fold the triangle to create the perpendicular bisector of each side of the triangle. What do you notice about the perpendicular bisectors? Make a conjecture about the perpendicular bisectors of the sides of a triangle.

4. Theorems Theorem 5-6 –The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 5-7 –The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

5. A little vocab… Concurrency –point where two or more segments meet Circumcenter of a triangle –point of concurrency of the perpendicular bisectors of a triangle –we can draw a circle using this point as our center that will pass through all vertices of the triangle (called circumscribed)

6. Circumscribed Circle see sketchpadsketchpad

7. Example 1 Find the center of the circle that you can circumscribe about ΔOPS

8. More Vocab… Incenter of a triangle –point of concurrency of the angle bisectors of a triangle inscribed –a circle drawn inside the triangle, using the incenter

9. Inscribed Circle see sketchpad #2sketchpad #2

10. Medians median of a triangle –segment whose endpoints are a vertex and the midpoint of the opposite side

11. Centroid point of concurrency of the medians this is the center of gravity of a triangle –it will balance at this point! see sketchpad #3sketchpad #3

12. Theorem 5-8 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.

13. Example 3 In ΔABC, D is the centroid and DE = 6. Find BE.

14. Altitudes the perpendicular segment from a vertex to the line containing the opposite side can be inside or outside the triangle

15. Example 4 Is ST a median, an altitude, or neither? Is UW a median, an altitude, or neither?

16. Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent. this point is called the orthocenterorthocenter

17. Recap Perpendicular Bisector: Angle Bisector: Median: Altitude:

18. Recap circumcenter: incenter: centroid: orthocenter:

19. Homework P. 275: 3-7, 11-16, 19-22