The co-ordinates of the vertex A of the ∆ ABC are (2, 5); if the centroid of the triangle is at (-2, 1), find the co-ordinates of the mid-point of the side BC.
The co-ordinates of the vertex A of the ∆ ABC are (2, 5); if the centroid of the triangle is at (-2, 1), find the co-ordinates of the mid-point of the side BC.
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Step-by-step explanation:
Let P(1, 1), Q(2, -3), R(3, 4) be the mid-points of sides AB, BC and CA respectively of triangle ABC. Let A(x
1
,y
1
),B(x
2
,y
2
)andC(x
3
,y
3
) be the vertices of triangle. ABC. Then,
P is the mid-point of BC
⇒
2
x
1
+x
2
=1,
2
y
1
+y
2
=1
⇒x
1
+x
2
=2andy
1
+y
2
=2....(i)
Q is the mid-point of BC
⇒
2
x
2
+x
3
=2,
2
y
2
+y
3
=−3
⇒x
2
+x
3
=4andy
2
+y
3
=−6....(ii)
R is the mid-point of AC
⇒
2
x
1
+x
3
=3and
2
y
1
+y
3
=4
⇒x
1
+x
3
=6andy
1
+y
3
=8...(iii)
From (i), (ii), and (iii), we get
x
1
+x
2
+x
2
+x
3
+x
1
+x
3
=2+4+6
and,y
1
+y
2
+y
2
+y
3
+y
1
+y
3
=2−6+8
⇒x
1
+x
2
+x
3
=6andy
1
+y
2
+y
3
=2...(iv)
The coordinates of the centroid of △ABC are
(
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)=(
3
6
,
3
2
)=(2,
3
2
) [Using (iv)]