the centroid of the triangle formed by the mid points of the side of the triangle with it's vertices (2,2)(0,2)(-2,2) is
Answers
Answer:
Step-by-step explanation:
Coordinates of the mid-point of A and B are
(
2
x
1
+x
2
,
2
y
1
+y
2
)
⟹
2
x
1
+x
2
=4,
2
y
1
+y
2
=2
∴x
1
+x
2
=8,y
1
+y
2
=4 (1)
Coordinates of mid-point of B and C are
2
x
2
+x
3
=3→x
2
+x
3
=6, (2)
2
y
2
+y
3
=3→y
2
+y
3
=6
Coordinates of mid-point of C and A are
2
x
1
+x
3
=2→x
1
+x
3
=4, (3)
2
y
1
+y
3
=2→y
1
+y
3
=4
Add the equation (1),(2) and (3)
2(x
1
+x
2
+x
3
)=182(y
1
+y
2
+y
3
)=14
⇒x
1
+x
2
+x
3
=9;x
1
+x
2
+x
3
=7
Given that x
1
+x
2
=8andy
1
+y
2
=4
8+x
3
=9⟹x
3
=1
4+y
3
=7⟹y
3
=3
x
1
+x
3
=4⟹x
1
=3
y
1
+y
3
=4⟹y
1
=1
∴x
2
=5andy
2
=3
∴ The vertices of△le A(3,1) ;B(5,3) ;C(1,3)
Centroid of triangle (
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
=(
3
9
,
3
7
)=(3,
3
7
)Coordinates of the mid-point of A and B are
(
2
x
1
+x
2
,
2
y
1
+y
2
)
⟹
2
x
1
+x
2
=4,
2
y
1
+y
2
=2
∴x
1
+x
2
=8,y
1
+y
2
=4 (1)
Coordinates of mid-point of B and C are
2
x
2
+x
3
=3→x
2
+x
3
=6, (2)
2
y
2
+y
3
=3→y
2
+y
3
=6
Coordinates of mid-point of C and A are
2
x
1
+x
3
=2→x
1
+x
3
=4, (3)
2
y
1
+y
3
=2→y
1
+y
3
=4
Add the equation (1),(2) and (3)
2(x
1
+x
2
+x
3
)=182(y
1
+y
2
+y
3
)=14
⇒x
1
+x
2
+x
3
=9;x
1
+x
2
+x
3
=7
Given that x
1
+x
2
=8andy
1
+y
2
=4
8+x
3
=9⟹x
3
=1
4+y
3
=7⟹y
3
=3
x
1
+x
3
=4⟹x
1
=3
y
1
+y
3
=4⟹y
1
=1
∴x
2
=5andy
2
=3
∴ The vertices of△le A(3,1) ;B(5,3) ;C(1,3)
Centroid of triangle (
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
=(
3
9
,
3
7
)=(3,
3
7
)Coordinates of the mid-point of A and B are
(
2
x
1
+x
2
,
2
y
1
+y
2
)
⟹
2
x
1
+x
2
=4,
2
y
1
+y
2
=2
∴x
1
+x
2
=8,y
1
+y
2
=4 (1)
Coordinates of mid-point of B and C are
2
x
2
+x
3
=3→x
2
+x
3
=6, (2)
2
y
2
+y
3
=3→y
2
+y
3
=6
Coordinates of mid-point of C and A are
2
x
1
+x
3
=2→x
1
+x
3
=4, (3)
2
y
1
+y
3
=2→y
1
+y
3
=4
Add the equation (1),(2) and (3)
2(x
1
+x
2
+x
3
)=182(y
1
+y
2
+y
3
)=14
⇒x
1
+x
2
+x
3
=9;x
1
+x
2
+x
3
=7
Given that x
1
+x
2
=8andy
1
+y
2
=4
8+x
3
=9⟹x
3
=1
4+y
3
=7⟹y
3
=3
x
1
+x
3
=4⟹x
1
=3
y
1
+y
3
=4⟹y
1
=1
∴x
2
=5andy
2
=3
∴ The vertices of△le A(3,1) ;B(5,3) ;C(1,3)
Centroid of triangle (
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
=(
3
9
,
3
7
)=(3,
3
7
)vCoordinates of the mid-point of A and B are
(
2
x
1
+x
2
,
2
y
1
+y
2
)
⟹
2
x
1
+x
2
=4,
2
y
1
+y
2
=2
∴x
1
+x
2
=8,y
1
+y
2
=4 (1)
Coordinates of mid-point of B and C are
2
x
2
+x
3
=3→x
2
+x
3
=6, (2)
2
y
2
+y
3
=3→y
2
+y
3
=6
Coordinates of mid-point of C and A are
2
x
1
+x
3
=2→x
1
+x
3
=4, (3)
2
y
1
+y
3
=2→y
1
+y
3
=4
Add the equation (1),(2) and (3)
2(x
1
+x
2
+x
3
)=182(y
1
+y
2
+y
3
)=14
⇒x
1
+x
2
+x
3
=9;x
1
+x
2
+x
3
=7
Given that x
1
+x
2
=8andy
1
+y
2
=4
8+x
3
=9⟹x
3
=1
4+y
3
=7⟹y
3
=3
x
1
+x
3
=4⟹x
1
=3
y
1
+y
3
=4⟹y
1
=1
∴x
2
=5andy
2
=3
∴ The vertices of△le A(3,1) ;B(5,3) ;C(1,3)
Centroid of triangle (
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
=(
3
9
,
3
7
)=(3,
3
7
)Coordinates of the mid-point of A and B are
(
2
x
1
+x
2
,
2
y
1
+y
2
)
⟹
2
x
1
+x
2
=4,
2
y
1
+y
2
=2
∴x
1
+x
2
=8,y
1
+y
2
=4 (1)
Coordinates of mid-point of B and C are
2
x
2
+x
3
=3→x
2
+x
3
=6, (2)
2
y
2
+y
3
=3→y
2
+y
3
=6
Coordinates of mid-point of C and A are
2
x
1
+x
3
=2→x
1
+x
3
=4, (3)
2
y
1
+y
3
=2→y
1
+y
3
=4
Add the equation (1),(2) and (3)
2(x
1
+x
2
+x
3
)=182(y
1
+y
2
+y
3
)=14
⇒x
1
+x
2
+x
3
=9;x
1
+x
2
+x
3
=7
Given that x
1
+x
2
=8andy
1
+y
2
=4
8+x
3
=9⟹x
3
=1
4+y
3
=7⟹y
3
=3
x
1
+x
3
=4⟹x
1
=3
y
1
+y
3
=4⟹y
1
=1
∴x
2
=5andy
2
=3
∴ The vertices of△le A(3,1) ;B(5,3) ;C(1,3)
Centroid of triangle (
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
=(
3
9
,
3
7
)=(3,
3
7
)
Step-by-step explanation:
hope this answer will help you
