Median and Altitude of a Triangle
Goal:
•To use properties of the medians of a triangle.
•To use properties of the altitudes of a triangle.
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Properties of Medians and Altitudes in Triangles, Lecture notes of Statistics
The concepts of medians and altitudes in triangles, their properties, and the points of concurrency. It covers the median of a triangle, median of an obtuse triangle, median of an acute triangle, median of a right triangle, altitude of a triangle, altitude of an acute triangle, altitude of a right triangle, and altitude of an obtuse triangle. The document also includes examples and a review of related concepts such as centroid, orthocenter, circumcenter, and incenter.
What you will learn
- Where does the median of a triangle intersect the triangle?
- What is the definition of an altitude in a triangle?
- What is the definition of a median in a triangle?
- What is the point of concurrency of the medians of a triangle called?
- Where does the altitude of a triangle intersect the triangle?
Typology: Lecture notes
2021/2022
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Median and Altitude of a Triangle
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.
Median
Vertex
Medians of a Triangle
A
C
B
P
F
E
D
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=
AF
CP= and BP=
3 3
CE BD
Example - Medians of a Triangle
A
C
B
P
F
E
D
.
5
P is the centroid of ABC
PF
Find AF and AP
5
Median of a Right Triangle
A
C
B
P
Point of concurrency “P” or centroid
The three medians of an obtuse,
acute, and a right triangle always
meet inside the triangle.
D
E
F
Altitude of a Triangle
A
C
B
altitude
Altitude of a triangle – the perpendicular segment
from the vertex to the opposite side or to the line
that contains the opposite side
Altitude of a Right Triangle
A
C
B
P
Point of concurrency “P” or orthocenter
The point of concurrency called the
orthocenter lies on the triangle.
The two legs are the altitudes
Altitude of an Obtuse Triangle
The point of concurrency of the three
altitudes is called the orthocenter
A
C
B
P
The point of concurrency lies
outside the triangle.
Example
Determine if EG is a perpendicular bisector, and angle bisector, a median,
or an altitude of triangle DEF given that:
a. DG FG
b. EG DF
c. DEG FEG
d. EG DF and DG FG
E
D
G
F
e. DEG FGE
Review
Properties / Points of Concurrency