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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.

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Bisectors medians altitudes

Bisectors, Medians, Altitudes

Chapter 5 Section 1

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


Points of concurrency
Points of Concurrency continually fearing you will make one. -- Elbert Hubbard

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
Draw the Perpendicular Bisectors continually fearing you will make one. -- Elbert Hubbard

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
Draw the Angle Bisectors continually fearing you will make one. -- Elbert Hubbard

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?


Draw the median of the triangle
Draw the Median of the Triangle continually fearing you will make one. -- Elbert Hubbard

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


Centroid is the point of balance
Centroid is the point of balance continually fearing you will make one. -- Elbert Hubbard


Centroid theorem
Centroid Theorem continually fearing you will make one. -- Elbert Hubbard

How does it work?

9

1/3

15

y

2/3

x


Centroid theorem1
Centroid Theorem continually fearing you will make one. -- Elbert Hubbard


Draw the altitudes of the triangle
Draw the Altitudes of the Triangle continually fearing you will make one. -- Elbert Hubbard

Their point of concurrency is called the orthocenter

Extend the line segments until they intersect


Coordinate geometry
Coordinate Geometry continually fearing you will make one. -- Elbert Hubbard

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 concurrency1
Points of Concurrency continually fearing you will make one. -- Elbert Hubbard

  • 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


Homework
Homework continually fearing you will make one. -- Elbert Hubbard

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


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