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.