Chapter 5: Relationships within Triangles

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 sketchpad

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 #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 #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 orthocenter

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

18. Recap • circumcenter: • incenter: • centroid: • orthocenter:

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