MPM2D
U02L07
The Centroid of a Triangle
The centroid of a triangle is the point of intersection of the three medians of the triangle. A median is a line segment which joins a vertex to the midpoint of the opposite side. The centroid represents the centre of gravity of the balance point of a triangle.
To find the coordinates of the centroid of a triangle we must find the equations of the medians of each side of the triangle.
• Find the midpoint of each side of the triangle
• Find the slope between the vertex and the midpoint of each side of the triangle
• Using the midpoint or the vertex and the slopes, determine the equation of the medians for each side of the triangle
• Using substitution or elimination with two of the median equations, determine the point of intersection
• Check this point in the third equation
Triangle ABC has vertices A (3,4) , B (-5,2) and C (1,-4). Determine
an equation (in standard form) for CD, the median from C to AB.
Example # 1:
y
6
5
A (3 ,
4
D
(­5 , 2)
B
4)
(­1 , 3)3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
­2
­3
­4
­5
­6
(1 ,
­4)
C
3
4
5
6
MPM2D
U02L07
Triangle PQR has vertices P (0,0) , Q (0,6) and R (12,0) . Determine
the coordinates of the centroid of triangle PQR.
Example # 2:
y
8
7
Q
(0 ,
6 6)
5
4
E
(0 ,
3)
D3
(6 ,
3)
2
R
1
P
(0 ,
0)
­2 ­1 0
­1
1
(6 ,
0)
2
3
4
5
6
F
(12 ,
0)
7
8
x
9 10 11 12 13
­2
Equation of the Median from vertex P
Equation of the Median from vertex R
Using substitution or elimination determine the
point of intersection of two of the medians.
Equation of the Median from vertex Q
Check in equation (2)
Sub. y = 2 into (1)
Therefore the coordinates of the centroid of triangle PQR is (4,2).
Triangle XYZ has vertices at X (0,0) , Y (4,4) and Z (8,-4).
Determine the coordinates of the centroid of triangle XYZ.
Example # 3:
y
6
5
Y (4 ,
4
4)
3
D
2
(2 ,
2)
1
X
­2
­1
0
­1
E
(0 ,
0)
1
2
3
4
­4
5
6
7
8
9
10
(4 ,
­2)
­2
­3
x
(6 ,
0)
F
(8 ,
­4)
Z
­5
­6
Homework: p. 195 # 8 , 9 , 12