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 BisectorsExtend the line segments

until they intersectTheir 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 GeometryThe 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. orthocenter

Geogebra\Geog_orthocenter.ggb

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)