Verify the following:

(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.


Let A(–1, 2, 1), B(1, –2, 5), C(4, –7, 8) & D(2, –3, 4)


ABCD can be vertices of parallelogram only if opposite sides are equal.


i.e. AB = CD & BC = AD


Calculating AB


A ≡ (– 1, 2, 1) and B ≡ (1, – 2, 5)


Distance AB


Here,


x1 = – 1, y1 = 2, z1 = 1


x2 = 1, y2 = – 2, z2 = 5


Distance AB






Calculating BC


B ≡ (1, – 2, 5) and C ≡ (4, – 7, 8)


Distance BC


Here,


x1 = 1, y1 = – 2, z1 = 5


x2 = 4, y2 = – 7, z2 = 8


Distance BC





Calculating CD


C ≡ (4, – 7, 8) and D ≡ (2, – 3, 4)


Distance CD


Here,


x1 = 4, y1 = – 7, z1 = 8


x2 = 2, y2 = – 3, z2 = 4


Distance CD






Calculating DA


D ≡ (2, – 3, 4) and A ≡ (– 1, 2, 1)


Distance DA


Here,


x1 = 2, y1 = – 3, z1 = 4


x2 = – 1, y2 = 2, z2 = 1


Distance DA





Since AB = CD & BC = DA


So, In ABCD both pairs of opposite sides are equal.


Thus, ABCD is a parallelogram.


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