Bisectors medians altitudes
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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

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Bisectors medians altitudes

  • 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

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

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

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?


Draw the median of the triangle

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


Centroid is the point of balance

Centroid is the point of balance


Centroid theorem

Centroid Theorem

How does it work?

9

1/3

15

y

2/3

x


Centroid theorem1

Centroid Theorem


Draw the altitudes of the triangle

Draw the Altitudes of the Triangle

Their point of concurrency is called the orthocenter

Extend the line segments until they intersect


Coordinate geometry

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 concurrency1

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


Homework

Homework

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


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