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 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

Draw the Perpendicular Bisectors

Extend the line segments Their point of until they intersect concurrency is called the circumcenter

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

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

The Centroid is the point of balance of any triangle Centroid is the point of balance Centroid Theorem

How does it work? 9 1 15 /3 2/ y 3 x Centroid Theorem Draw the Altitudes of the Triangle

Extend the line Their point of segments until concurrency is they intersect called the orthocenter 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. Points of Concurrency

Questions: 1. Will the P.O.C. always be inside the Hyperlink to Geogebra Figures triangle? 1. circumcenter Geogebra\Geog_Circumcenter.ggb 2. If you distort the Triangle, do the Special Segments change? 2. incenter Geogebra\Geog_Incenter.ggb 3. centroidGeogebra\Geog_centroid.ggb 3. Can you move the special segments by 4. orthocenterGeogebra\Geog_orthocenter.g themselves? gb Homework

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