using vectors prove that the diagonals of a parallelogram bisect each other
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We draw a parallelogram with one diagonal coincident to x-axis and the intersect of two diagonals is on origin. One diagonal into divided into
a
and m
a
, the other is
b
and n
b
.
Now,
⇒
a
+
l
=
b
and
l
+m
a
=n
b
(where m and n are scalars)
⇒
a
-
b
=m
a
-n
b
⇒(m−1)
a
=(n−1)
b
So,
a
=(
∣a∣
0
)
b
=(
∣b∣cos θ
∣b∣sin θ
)
y direction:
0=(n−1)∣b∣sin θ
In the parallelogram, θ>0 and ∣b∣>0
∴n=1 and
b
=n
b
Hence, diagonal b is bisected.
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