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Byju's Answer
Standard IX
Mathematics
The Mid-Point Theorem
If Aa⃗,Bb⃗,...
Question
If
A
(
→
a
)
,
B
(
→
b
)
,
C
(
→
c
)
are vertices of
Δ
A
B
C
of then coordinates of centroid are
A
→
g
=
→
a
+
→
b
+
→
c
3
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B
→
g
=
→
a
+
→
b
+
→
c
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C
→
g
=
→
a
−
→
b
−
→
c
3
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D
→
g
=
→
a
−
→
b
+
→
c
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Solution
The correct option is
D
→
g
=
→
a
+
→
b
+
→
c
3
Given vertices
→
a
,
→
b
,
→
c
centroid(g) is the point from which the distance between centroid and vertices are equal
distance
A
G
A
G
=
→
g
−
→
a
-------(1)
distance
B
G
B
G
=
→
g
−
→
B
-------(2)
distance
C
G
C
G
=
→
g
−
→
c
-------(3)
then
A
G
+
B
G
+
C
G
→
g
−
→
a
+
→
g
−
→
b
+
→
g
−
→
c
=
0
3
→
g
−
(
→
a
+
→
b
+
→
c
)
=
0
3
→
g
=
→
a
+
→
b
+
→
c
→
g
=
→
a
+
→
b
+
→
c
3
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0
Similar questions
Q.
Let
→
A
=
→
b
×
→
c
,
→
B
=
→
c
×
→
a
,
→
C
=
→
a
×
→
b
, then the vectors
→
A
×
(
→
B
×
→
C
)
,
→
B
×
(
→
C
×
→
A
)
, and
→
C
×
(
→
A
×
→
B
)
are
Q.
If
→
a
,
→
b
,
→
c
are positive vectors of vertices
A
,
B
,
C
of
Δ
A
B
C
. If
→
r
is position vector of a point
P
such that
(
|
→
b
−
→
c
|
+
|
→
c
−
→
a
|
+
|
→
a
−
→
b
|
)
→
r
=
|
→
b
−
→
c
|
→
a
+
|
→
c
−
→
a
|
→
b
+
∣
∣
→
a
−
→
b
∣
∣
→
c
then the point
P
always
Q.
Statement-
1
: If vertices of a triangle are
A
(
→
a
)
,
B
(
→
b
)
,
C
(
→
c
)
then length of altitude through
A
is
|
→
a
×
→
b
+
→
b
×
→
c
+
→
c
×
→
a
|
|
→
b
−
→
c
|
Statement-
2
: Area of triangle is
Δ
=
1
2
(Base) (Height)
=
1
2
|
→
A
B
×
→
A
C
|
Q.
If
→
a
,
→
b
,
→
c
are unit vectors such that
→
a
+
→
b
+
→
c
=
→
0
and
(
→
a
,
→
b
)
=
π
3
, then
∣
∣
→
a
×
→
b
∣
∣
+
∣
∣
→
b
×
→
c
∣
∣
+
|
→
c
×
→
a
|
=
Q.
If
→
a
,
→
b
,
→
c
are three vectors such that
→
a
×
→
b
=
→
c
,
→
b
×
→
c
=
→
a
,
→
c
×
→
a
=
→
b
then prove that
|
→
a
|
=
|
→
b
|
=
|
→
c
|
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