COORDINATE GEOMETRY
Centroid formula
Centroid
Point of concurrence of medians
Consider ΔABC
A
B
C
(x1, y1)
(x2, y2)
(x3, y3)
Q
R
P
Let us draw medians AP, BQ and CR
Segment joining vertex and midpoint of opposite side
∴Coordinates of G are
=
x1
+
x2
3
,
y1
+
y2
3
+
x3
+
y3
G
A (x1, y1)
B (x2, y2)
C (x3, y3)
B
C
A
G
(3, 2)
(–2, 1)
(x, y)
5
3
1
3
–
,
Sol.
∴
By Centroid formula,
=
3
–
2
3
,
2
+
1
3
x1 = 3,
y1 = 2
x2 = –2,
y2 = 1
A(3, 2), B(–2, 1), G
5
3
1
3
–
,
Let coordinates of C are (x, y)
x3 = x,
y3 = y
∴Coordinates of G
=
x1
+
x2
3
,
y1
+
y2
3
+
x3
+
y3
+
x
+
y
5
3
1
3
–
,
=
1
3
,
3
3
+
x
+
y
5
3
1
3
–
,
Let the co-ordinates of A be (x1, y1)
Let the co-ordinates of B be (x2, y2)
Let the co-ordinates of C be (x3, y3)
5
3
=
∴
1 + x
3
–1
3
=
∴
3 + y
3
Centroid Formula
,
+
x1
x2
+
x3
3
+
y1
y2
3
+
y3
B
C
A
G
(3, 2)
(–2, 1)
(x, y)
5
3
1
3
–
,
Sol.
5
3
=
∴
1 + x
3
–1
3
=
∴
3 + y
3
5
∴
=
1
+
x
5
∴
–
1
=
x
∴
x
=
4
–1
∴
=
3
+
y
–1
∴
–
3
=
y
∴
y
=
–4
∴
The coordinates of C are (4, –4)