Bisectors, Medians, Altitudes

Chapter 5 Section 1

Learning Goal: Understand and Draw the concurrent points of a Triangle

The greatest mistake you can make in life is to be continually fearing you will

make one. -- Elbert Hubbard

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 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?

Draw the Angle Bisectors

Extend the line segments until they intersect

Their point of concurrency is called

the incenter

What do you notice?

Draw a circle with center at the incenter and the distance from the incenter to the side

as the radius of the circle

Draw the Median of the Triangle

Extend the line segments until they intersect

Their point of concurrency is

called the centroid

The Centroid is the point of balance of any triangle

Centroid is the point of balance

Centroid Theorem

2/3

1/3

How does it work? 9

x

15

y

Centroid Theorem

Draw the Altitudes of the Triangle

Extend the line segments until they intersect

Their point of concurrency is

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

Hyperlink to Geogebra Figures

1. circumcenter Geogebra\Geog_Circumcenter.ggb

2. incenter Geogebra\Geog_Incenter.ggb

3. centroidGeogebra\Geog_centroid.ggb

4. orthocenterGeogebra\Geog_orthocenter.g

gb

Questions:

1. Will the P.O.C. always be inside the triangle?

2. If you distort the Triangle, do the Special Segments change?

3. Can you move the special segments by themselves?

Homework

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