Bisectors, Medians, Altitudes

1. The greatest mistake you can make in life is to be continually fearing you will make one. -- Elbert Hubbard Bisectors, Medians, Altitudes Chapter 5 Section 1 Learning Goal: Understand and Draw the concurrent points of a Triangle

2. Points of Concurrency When three or more lines intersect at a common point, the lines are called Concurrent Lines. Their point of intersection is called the point of concurrency. Concurrent Lines Non-Concurrent Lines

3. Draw the Perpendicular Bisectors Extend the line segments until they intersect Their point of concurrency is called the circumcenter Draw a circle with center at the circumcenter and a vertex as the radius of the circle What do you notice?

4. Draw the Angle Bisectors Extend the line segments until they intersect Their point of concurrency is called the incenter Draw a circle with center at the incenter and the distance from the incenter to the side as the radius of the circle What do you notice?

5. Draw the Median of the Triangle Their point of concurrency is called the centroid Extend the line segments until they intersect The Centroid is the point of balance of any triangle

6. Centroid is the point of balance

7. Centroid Theorem How does it work? 9 1/3 15 y 2/3 x

8. Centroid Theorem

9. Draw the Altitudes of the Triangle Their point of concurrency is called the orthocenter Extend the line segments until they intersect

10. Coordinate Geometry The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC.

11. Points of Concurrency • Questions: • Will the P.O.C. always be inside the triangle? • If you distort the Triangle, do the Special Segments change? • Can you move the special segments by themselves? • Hyperlink to Geogebra Figures • circumcenter Geogebra\Geog_Circumcenter.ggb • incenter Geogebra\Geog_Incenter.ggb • centroidGeogebra\Geog_centroid.ggb • orthocenterGeogebra\Geog_orthocenter.ggb

12. Homework • Pages 275 – 277; #16, 27, 32 – 35 (all), 38, 42, and 43. (9 problems)