prove that the centroid of the triangle whose vertices are A (1,-6) ,B(-5,2),C(1,-2) obtained by joining the mid point of the sides of a triangle is the same as the centroid of the original triangle
Answers
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)]