prove by vector method the line joining the mid points of consecutive sides of a quadrilateral is a parallelogram.
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prove by vector method the line joining the mid points of consecutive sides of a quadrilateral is a parallelogram.
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Let ABCD be a quadrilateral and M,N,O,P be the mid points of the sides AB,BC,CD,DA respectively.
Position vectors of M,N,O,P are
2
a
+
b
,
2
b
+
c
,
2
c
+
d
,
2
d
+
a
respectively.
If we show that
MN
=
PO
MP
=
NO
, then it means MNOP is a parallelogram.
MN
=
2
b
+
c
−
2
a
+
b
=
2
c
−
a
PO
=
2
c
+
d
−
2
d
+
a
=
2
c
−
a
∴
MN
=
PO
⇒
MN
∥
PO
Similarly, we can prove that
MP
=
NO
and
MP
∥
NO
Hence, MNOP is a parallelogram.
Position vectors of M,N,O,P are
2
a
+
b
,
2
b
+
c
,
2
c
+
d
,
2
d
+
a
respectively.
If we show that
MN
=
PO
MP
=
NO
, then it means MNOP is a parallelogram.
MN
=
2
b
+
c
−
2
a
+
b
=
2
c
−
a
PO
=
2
c
+
d
−
2
d
+
a
=
2
c
−
a
∴
MN
=
PO
⇒
MN
∥
PO
Similarly, we can prove that
MP
=
NO
and
MP
∥
NO
Hence, MNOP is a parallelogram.
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