Geometry
Lesson 5 – 2
Medians and Altitudes of
Triangles
Objective:
Identify and use medians in triangles.
Identify and use attitudes in triangles.
Median
Median of a triangle

A segment with endpoints at a vertex of a
triangle and the midpoint of the opposite
side.
Centroid
Centroid

The point of concurrency of the medians of a
triangle.
Centroid Theorem

The medians of a triangle intersect at a point
called the centroid that is two thirds of the
distance from each vertex to the midpoint of
the opposite side.
Centroid…
2x
x
PK + AP = AK
PK + 2(PK) = AK
PK = 5
Find AP
10
BP = 12
Find PL. 6
JC = 15
Find JP.
5
In triangle ABC, Q is the centroid and
BE = 9
Find BQ
2
BQ  BE 
3
2
 9 
3
6
Find QE
1
QE  BE 
3
1
 9 
3
=3
OR
BQ = 2(QE)
6 = 2(QE)
3 = QE
In triangle ABC, Q is the centroid and
FC = 14
Find FQ
1
FQ  ( FC )
3
1
 (15)
3
=5
Find QC
QC = 2(FQ)
QC = 2(5)
QC = 10
In triangle JKL, PT = 2. Find KP.
How do you know that P is the
centroid?
KP = 2(PT) = 2(2) = 4
OR
2
KP  KT 
3
2
KP  KP  PT 
3
2
KP  KP  2 
3
2
4
KP  ( KP) 
3
3
1
4
KP  
3
3
KP = 4
In triangle JKL, RP = 3.5 and JP = 9
Find PL
PL = 2(RP)
= 2(3.5)
=7
Find PS
JP = 2(PS)
9 = 2(PS)
PS = 4.5
A performance artist plans to balance
triangular pieces of metal during her
next act. When one such triangle is
placed on the coordinate plane, its
vertices are located at (1, 10) (5, 0) and
(9,5). What are the coordinates of the
point where the artist should support the
triangle so that it will balance.
The balance point of a triangle is the centroid.
Graph the points.
Hint: To make it easier look for a vertical or
horizontal line between a midpoint of a side and
vertex.
Find the midpoint of the side(s) that could make a
vertical or horizontal line.
Find the midpoint of AB.
 1  5 10  0 
,
Midpoint of AB = 

2 
 2
= (3, 5)
Let P be the Centroid, where would it be?
From the vertex to the centroid is 2/3 of the whole.
2
CP  (CD)
3
2
CP  (6)
3
CP  4
Count over from C 4 units and that is P
Centroid: (5, 5)
A second triangle has vertices (0,4), (6,
11.5), and (12,1). What are the
coordinates of the point where the artist
should support the triangle so that it will
balance? Explain your reasoning.
Centroid: (6, 5.5)
Altitude
Altitude of a triangle

A perpendicular segment from a vertex to
the side opposite that vertex.
Draw a right
triangle and identify
all the altitudes.
Orthocenter
Orthocenter

The lines containing the altitudes of a
triangle are concurrent, intersecting at a
point called the orthocenter.
Find the orthocenter
The vertices of triangle FGH are F(-2, 4),
G(4,4), and H(1, -2). Find the coordinates of
the orthocenter of triangle FGH.
Graph the points.
Cont…
Find an equation from F to GH.
Slope of GH. m = 2
New equation is perpendicular to segment GH.
Point F (-2, 4) m = -1/2
y = mx + b
 1
4     2  b
 2
3=b
1
y   x3
2
Cont…
Find an equation from G to FH.
Slope of segment FH m = -2
New equation is perpendicular to segment FH.
Point G (4, 4) m = 1/2
1
4  4  b
2
2=b
1
y  x2
2
Cont…
The orthocenter can be found at the
intersection of our 2 new equations.
How can we find the orthocenter?
If the orthocenter lies on an exact point of the graph
use the graph to name. If it does not lie on a point
use systems of equations to find the orthocenter.
System of equations:
1
y   x3
2
1
y  x2
2
Cont…
1
y   x3
2
1
y  x2
2
1
1
 x3 x2
2
2
3 x2
1=x
1
y  1  2
2
y = 2.5
By substitution.
Orthocenter
(1, 2.5)
Summary
Perpendicular bisector
Summary
Angle bisector
Summary
Median
Summary
Altitude
Homework
Pg. 337 1 – 10 all, 12 - 20 E,
27 – 30 all, 48 – 54 E