1. Section 5-3: Concurrent Lines, Medians, and Altitudes March 6, 2012

2. Warm-up Warm-up:  Practice 5-1: 4-12  Practice 5-2: 6-21

3. Warm-up Practice 5-1: 4-12

4. Warm-up Practice 5-1: 4-12

5. Warm-up Practice 5-1: 4-12

6. Questions on Homework?

7. Section 5-3: Concurrent Lines, Medians, and Altitudes  Objectives: Today you will learn to identify properties of perpendicular and angle bisectors, and medians and altitudes of a triangle.

8. Section 5-3: Bisectors of Triangles Recall:  Perpendicular Bisector: line or segment that is perpendicular to a segment at its midpoint.  Angle Bisector: ray, line or segment that divides an angle into two congruent angles.

9. Section 5-3: Median and Altitude of Triangles  Median: segment whose endpoints are a vertex and the midpoint of the opposite side  Altitude: perpendicular segment from a vertex to the line containing the opposite side  Point of Concurrency: the point at which three or more lines intersect

10. Section 5-3: Perpendicular Bisectors and Acute Triangles 1.Label one paper “Perpendicular Bisector.” 2.Draw an acute triangle on the paper. 3.Fold the paper so that one side is exactly on top of itself; so it is cut in half. That is the perpendicular bisector. 4.Repeat with remaining two sides. 5.Where do the perpendicular bisectors intersect?

11. Section 5-3: Angle Bisectors and Acute Triangles 1.Label the other paper “Angle Bisector.” 2.Draw an acute triangle on the paper. 3.Fold the paper so that one angle is cut in half. That is the angle bisector. 4.Repeat with remaining two angles. 5.Where do the angle bisectors intersect?

12. Section 5-3: Exploration with Geogebra http://www.geogebra.org

13. Section 5-3: Theorems (p. 257-259) 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. 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. Theorem 5-9: The lines that contain the altitudes of a triangle are concurrent.

14. Section 5-3: Review Perpendicular Bisectors Angle Bisectors Medians Altitudes

15. Section 5-3: Hospital Location Boise, ID; Helena, MT; and Salt Lake City, UT, three large cities in the US, want to build a new modern Hospital that they can share. But where should it be built?

16. Section 5-3: Hospital Location

17. Helena (1, 6) Salt Lake City (2, -5) Boise (-4,0)

18. Section 5-3: Hospital Location Pt of concurrency (1.5, 0.5)

19. Wrap-up  Today you learned to identify properties of perpendicular and angle bisectors, and medians and altitudes of a triangle.  Tomorrow you’ll learn about Indirect Reasoning  Quiz on sections 5-1 to 5-3 on Thursday!  Homework p. 259-262: 1-5, 8-16, 19-22, 27-29, 33, 34