Transcript Document

Median and Altitude of a Triangle
Sec 5.3
Goal:
To use properties of the medians of a
triangle.
To use properties of the altitudes of a
triangle.
Median of a Triangle
Median of a Triangle – a segment whose endpoints are the vertex
of a triangle and the midpoint of the opposite side.
Vertex
Median
Median of an Obtuse Triangle
A
Point of concurrency “P” or centroid
D
E
P
B

F
C
Medians of a Triangle
Theorem 5.7
The medians of a triangle intersect at a point that is two-thirds of
the distance from each vertex to the midpoint of the opposite side.
If P is the centroid of ABC, then
AP= 2 AF
3
A
D
CP= 2 CE and BP= 2 BD
3
3

P
E
B
F
C
Example - Medians of a Triangle
P is the centroid of
PF  5
Find AF and AP
ABC.
A
D
E
B

P 5
F
C
Median of an Acute Triangle
A
Point of concurrency “P” or centroid
E

B
P
F
D
C
Median of a Right Triangle
A
E
B
P

F
D
Point of concurrency “P” or centroid
C
The three medians of an obtuse,
acute, and a right triangle always
meet inside the triangle.
Altitude of a Triangle
Altitude of a triangle – the perpendicular segment
from the vertex to the opposite side or to the line
that contains the opposite side
A
altitude
B
C
Altitude of an Acute Triangle
A
Point of concurrency “P” or orthocenter

P
B
C
The point of concurrency called the
orthocenter lies inside the triangle.
Altitude of a Right Triangle
The two legs are the altitudes
A
B
The point of concurrency called the
orthocenter lies on the triangle.

P
C
Point of concurrency “P” or orthocenter
Altitude of an Obtuse Triangle
A
B
P

The point of concurrency lies
outside the triangle.
C
The point of concurrency of the three
altitudes is called the orthocenter
Altitudes of a Triangle
Theorem 5.8
The lines containing the altitudes of a triangle are concurrent.
A
F
E
C
B
D
P 
If AE, BF , and CD are the altitudes of ABC,
then the lines AE, BF , and CD intersect at P.
Example
Determine if EG is a perpendicular bisector, and angle bisector, a
median, or an altitude of triangle DEF given that:
a. DG  FG
E
b. EG  DF
c. DEG FEG
d. EG  DF and DG  FG
e. DEG  FGE
D
G
F
Review
Properties / Points of Concurrency
Median -- Centroid
Altitude -- Orthocenter
Perpendicular Bisector -- Circumcenter
Angle Bisector -- Incenter