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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
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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
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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?
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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
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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
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Centroid is the point of balance
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Centroid Theorem
2/3
1/3
How does it work? 9
x
15
y
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Centroid Theorem
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Draw the Altitudes of the Triangle
Extend the line segments until they intersect
Their point of concurrency is
called the orthocenter
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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.
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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?
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Homework
Pages 275 – 277; #16, 27, 32 – 35 (all), 38, 42, and 43. (9 problems)