Bisectors, Medians, Altitudes

1
Bisectors, Medians, Altitudes

• The greatest mistake you can make in life is to
  be continually fearing you will make one. --
  Elbert Hubbard

• Chapter 5 Section 1
• Learning Goal Understand and Draw the concurrent
  points of a Triangle

2
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
3
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?
4
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?
5
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
6
Centroid is the point of balance
7
Centroid Theorem
How does it work?
9
1/3
15
y
2/3
x
8
Centroid Theorem
9
Draw the Altitudes of the Triangle
Their point of concurrency is called the
orthocenter
Extend the line segments until they intersect
10
Coordinate Geometry
The vertices of ?ABC are A(2, 2), B(4, 4), and
C(1, 2). Find the coordinates of the orthocenter
of ?ABC.
11
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

12
Homework

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