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Byju's Answer
Standard XII
Mathematics
Addition of Vectors
In quadrilate...
Question
In quadrilateral ABCD, prove that
(i)
¯
¯¯¯¯¯¯
¯
A
B
+
¯
¯¯¯¯¯¯
¯
B
C
+
¯
¯¯¯¯¯¯¯
¯
A
D
+
¯
¯¯¯¯¯¯¯
¯
D
C
=
2
¯
¯¯¯¯¯¯
¯
A
C
(ii)
¯
¯¯¯¯¯¯¯
¯
A
D
+
¯
¯¯¯¯¯¯
¯
B
C
=
2
¯
¯¯¯¯¯¯¯¯¯
¯
M
N
, where M and N are the mid points of side AB and CD respectively.
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Solution
(
i
)
→
A
B
+
→
B
C
=
→
A
C
→
(
1
)
→
A
D
+
→
D
C
=
→
A
C
→
(
2
)
Adding (1) and (2) ,
→
A
B
+
→
B
C
+
→
A
D
+
→
D
C
=
2
→
A
C
(
i
i
)
→
M
N
+
→
N
C
=
→
M
C
,
→
M
B
+
→
B
C
=
→
M
C
→
M
N
+
→
N
C
=
→
M
B
+
→
B
C
→
(
1
)
→
M
A
+
→
A
D
=
→
M
D
,
→
M
N
+
→
N
D
=
→
M
D
→
M
A
+
→
A
D
=
→
M
N
+
→
N
D
→
(
2
)
From (1) ,
→
M
N
−
→
N
D
=
→
M
B
+
→
B
C
[
∵
→
N
C
=
−
→
N
D
]
→
M
N
−
→
M
B
=
→
N
D
+
→
B
C
→
M
N
+
→
M
A
=
→
N
D
+
→
B
C
[
∵
→
M
A
=
−
→
M
B
]
→
M
N
+
→
M
N
+
→
N
D
−
→
A
D
=
→
N
D
+
→
B
C
[From (2),
→
M
A
=
→
M
N
+
→
N
D
]
2
→
M
N
+
→
N
D
=
→
N
D
+
→
A
D
+
→
B
C
2
→
M
N
=
→
A
D
+
→
B
C
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0
Similar questions
Q.
ABCD is a kite where AB =AD and BC =CD .P,Q,R and S are the mid -points of the sides AB, BC, CD and AD respectively. Prove that PQRS is a rectangle.
Q.
A
B
C
D
is a trapezium in which
A
B
|
|
D
C
.
M
and
N
are mid-points of side
A
D
and side
B
C
respectively. If
A
B
=
12
c
m
and
M
N
=
14
c
m
, find
C
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Q.
□
ABCD is a trapezium in which AB || DC. M and N are mid-points of side AD and side BC respectively. If AB = 12 cm. and MN = 14 cm, find CD.