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Byju's Answer
Standard XII
Mathematics
Slope of a Line Connecting Two Points
If ABCD is a ...
Question
If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, prove that
→
A
B
+
→
A
D
+
→
C
B
+
→
C
D
=
4
→
E
F
.
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Solution
Let the position vectors of vertices A, B, C and D of quadrilateral ABCD be
→
a
,
→
b
,
→
c
and
→
d
respectively.
E is the mid point of AC
∴
Position vector of
E
=
1
2
(
→
a
+
→
c
)
F is the mid point of BD
∴
Position vector of
F
=
1
2
(
→
b
+
→
d
)
∴
→
E
F
=
position vector of F
=
position vector of
^
t
=
1
2
(
→
b
+
→
d
)
−
1
2
(
→
a
+
→
c
)
=
1
2
(
→
b
+
→
d
−
→
a
−
→
c
)
Now
→
A
B
+
→
A
D
+
→
C
B
+
→
C
D
=
(
→
b
⋅
→
a
)
+
(
→
d
−
→
c
)
+
(
→
d
−
→
c
)
=
2
(
→
b
+
→
d
−
→
a
−
→
c
)
=
2
×
2
→
E
F
=
4
→
E
F
.
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Similar questions
Q.
If
A
,
B
,
C
and
D
are any four points E and F are the middle points of
A
C
and
B
D
, respectively then prove that
→
C
B
+
→
C
D
+
→
A
D
+
→
A
B
=
4
→
E
F
.
Q.
If
M
and
N
are mid points of the diagonals
A
C
and
B
D
respectively of a quadrilateral
A
B
C
D
show that
→
A
B
+
→
A
D
+
→
C
B
+
→
C
D
=
4
→
M
N
.
Q.
If
A
,
B
,
C
,
D
be any four points and
E
and
F
be the mid-points of
A
C
and
B
D
, respectively, then
→
A
B
+
→
C
B
+
→
C
D
+
→
A
D
is equal to
Q.
If P and Q are the midpoints of the diagonals AC and BD respectively of a quadrilateral ABCD, prove that
¯
¯¯¯¯¯¯
¯
A
B
+
¯
¯¯¯¯¯¯¯
¯
A
D
+
¯
¯¯¯¯¯¯
¯
C
B
+
¯
¯¯¯¯¯¯¯
¯
C
D
=
4
¯
¯¯¯¯¯¯
¯
P
Q
.
Q.
In a quadrilateral
A
B
C
D
,
→
A
C
is the bisector of the
(
→
A
B
∧
→
A
D
)
which is
2
π
3
,
15
∣
∣
→
A
C
∣
∣
=
3
∣
∣
→
A
B
∣
∣
=
5
∣
∣
→
A
D
∣
∣
then
cos
(
→
B
A
∧
→
C
D
)
is :
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