1. A median of a triangle is a segment whose endpoints are
USING MEDIANS OF A TRIANGLE
A median of a triangle is a
segment whose endpoints are
a vertex of the triangle and the
midpoint of the opposite side. D

2. The three medians of a triangle are concurrent. The point
USING MEDIANS OF A TRIANGLE
The three medians of a triangle are concurrent. The point
of concurrency is called the centroid of the triangle. The
centroid is always inside the triangle.
acute triangle
right triangle
obtuse triangle
centroid
The medians of a triangle have a special concurrency property.

3. USING MEDIANS OF A TRIANGLE
THEOREM
THEOREM Concurrency of Medians of a Triangle
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 = AD 2 3 P
BP = BF 2 3
CP = CE 2 3

4. USING MEDIANS OF A TRIANGLE
The centroid of a triangle
can be used as its balancing point.
A triangular model of uniform thickness and density will balance at the centroid of the triangle.

5. P is the centroid of QRS shown below and PT = 5. Find RT and RP.
Using the Centroid of a Triangle
P is the centroid of QRS shown below and PT = 5.
Find RT and RP.
SOLUTION
Because P is the centroid,
RP = RT. 2 3
Then PT = RT – RP = RT 1 3
Substituting 5 for PT, 5 = RT, so RT = 15. 1 3
Then RP = RT = (15) = 10. 2 3
So, RP = 10 and RT = 15.

6. Finding the Centroid of a Triangle
Find the coordinates of the centroid of  JKL.
(5, 8)
SOLUTION J K L N P
(7, 10)
(3, 6)
(5, 2)
The centroid is two thirds of the
distance from each vertex to the midpoint of the opposite side.
Choose the median KN. Find the coordinates of N, the midpoint
of JL. 2 , = 10 16
= (5, 8)
The coordinates of N are

7. The coordinates of the centroid P are (5, 2 + 4), or (5, 6)
Finding the Centroid of a Triangle
Find the coordinates of the centroid of  JKL.
(5, 8)
SOLUTION J K L
(7, 10)
(3, 6)
(5, 2)
Find the distance from vertex K to midpoint N. The distance from K (5,2) to N(5,8) is 8 – 2, or 6 units. N
(5, 6) P
Determine the coordinates of the centroid, which is • 6, or 4 units up from vertex K along
the median KN. 2 3 M
The coordinates of the centroid P are (5, 2 + 4), or (5, 6)
[Yellow coordinates appear.]

8. USING ALTITUDES OF A TRIANGLE
An altitude of a triangle is the perpendicular segment
from a vertex to the opposite side or to the line that contains
the opposite side. An altitude can lie inside, on, or outside
the triangle.
Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called
the ortho center of the triangle.

9. Where is the ortho center of an acute triangle?
Drawing Altitudes and Orthocenters
Where is the ortho center of an acute triangle?
SOLUTION
Draw an example.
The three altitudes intersect at G,
a point inside the triangle.

10. Where is the orthocenter of a right triangle?
Drawing Altitudes and Orthocenters
Where is the orthocenter of a right triangle?
SOLUTION
The two legs, LM and KM,
are also altitudes. They intersect at the triangle’s right angle.
This implies that the orthocenter is on the triangle
at M, the vertex of the right angle of the triangle.

11. Where is the orthocenter of an obtuse triangle?
Drawing Altitudes and Orthocenters
Where is the orthocenter of an obtuse triangle?
SOLUTION
The three lines that contain the altitudes intersect at W,
a point outside the triangle.

12. USING ALTITUDES OF A TRIANGLE
THEOREM
THEOREM Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle
are concurrent.
If AE, BF, and CD are the altitudes
of ABC, then the lines AE, BF, and
CD intersect at some point H.