Medians and Altitudes of Triangles

1. MEDIANS AND ALTITUDES OF
A TRIANGLE
B
E
D
G
A F E

2. MEDIANS OF A TRIANGLE
A median of a triangle A
is a segments whose
endpoints are a vertex
of the triangle and the MEDIAN
midpoint of the
opposite side. In the
figure, in ∆ ABC ,
shown at the right, D
is the midpoint of side
BC . So, segment AD C
is a median of the D
B
triangle
2

3. CENTROIDS OF THE TRIANGLE
The three medians of a
triangle are
concurrent. This
means that they meet
at a point. The point
of concurrency is
called the CENTROID
OF THE TRIANGLE. CENTROID
The centroid, labeled P
P in the diagrams in
the next few slides
are ALWAYS inside
the triangle. acute triangle
3

4. CENTROIDS
P centroid P centroid
RIGHT TRIANGLE obtuse triangle
You see that the centroid is ALWAYS INSIDE THE TRIANGLE
4

5. T H E ORE M : C ON C URRE NC Y OF M E DI ANS OF A T RI A NG LE
The medians of a
triangle intersect at a B
point that is two thirds
of the distance from D
each vertex to the
midpoint of the
opposite side. E C
If P is the centroid of P
∆ABC, then F
AP = 2/3 AD,
BP = 2/3 BF, and A
CP = 2/3 CE
5

6. EXERCISE: USING THE CENTROID OF A
TRIANGLE
P is the centroid
of ∆QRS
shown below
R
and PT = 5.
Find RT and
RP. S
P
T
Q
6

7. EXERCISE: USING THE CENTROID OF A TRIANGLE
Because P is the
centroid. RP = 2/3 R
RT.
Then PT= RT – RP =
1/3 RT.
Substituting 5 for PT, S
5 = 1/3 RT, so P
RT = 15. T
Then RP = 2/3 RT = Q
2/3 (15) = 10
► So, RP = 10, and
RT = 15.
7

8. EXERCISE: FINDING THE CENTROID OF A TRIANGLE
J (7, 10)
The coordinates of N are: 10
3+7 , 6+10 = 10 , 16
2 2 2 2 8
N
Or (5, 8)
6 L (3, 6)
P
Find the distance from
vertex K to midpoint N. 4
The distance from K(5,
M
2) to N (5, 8) is 8-2 or 6
units. 2
K (5, 2)
8

9. EXERCISE: DRAWING ALTITUDES AND
ORTHOCENTERS
 Where is the orthocenter located in
each type of triangle?
a. Acute triangle
b. Right triangle
c. Obtuse triangle
9

10. ACUTE TRIANGLE - ORTHOCENTER
B
E
D
G
A F E
∆ABC is an acute triangle. The three altitudes
intersect at G, a point INSIDE the triangle.
10

11. RIGHT TRIANGLE - ORTHOCENTER
∆KLM is a right triangle.
The two
K legs, LM and KM, are also
altitudes.
J
They intersect at the
triangle’s right angle. This
implies that the orthocenter
M L is ON the triangle at M, the
vertex of the right angle of
thetriangle.
11

12. OBTUSE TRIANGLE - ORTHOCENTER
P
Z
W Y Q
X
R
∆YPR is an obtuse triangle. The three lines that contain the
altitudes intersect at W, a point that is OUTSIDE the
triangle.
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13. CONCURRENCY OF ALTITUDES OF A TRIANGLE
F B
H
A
E
D
C
The lines containing the altitudes of a triangle are
concurrent.
If AE, BF, and CD are altitudes of ∆ABC, then the
lines AE, BF, and CD intersect at some point H.
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