Question:
Three vertices of a parallelogram ABCD are A(3, - 1, 2), B (1, 2, - 4) and C (- 1, 1, 2). Find the coordinates of the fourth vertex.
Answer:
Let fourth vertex is D (x,y,z)
Now mid point of AC = {(3-1)/2, (-1+1)/2, (2+2)/2} = (1,0,2)
mid point of BD = {(x+1)/2, (y+2)/2, (z-4)/2}
Now
mid point of AC = mid point of BD
=> (1,0,2) = {(x+1)/2, (y+2)/2, (z-4)/2}
(x+1)/2 = 1
=> x+1 = 2
=> x = 2-1
=> x = 1
(y+2)/2 = 0
=> y+2 = 0
=> y = -2
(z-4)/2 = 2
=> z-4 = 2*2
=> z-4 = 4
=> z = 4+4
=> z = 8
So the point D is (1,-2,8)
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