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(B) Let the vertices of the parallelogram be $A(1, 2, 3)$,$B(-1, -2, -1)$,$C(2, 3, 2)$,and $D(x, y, z)$.In a parallelogram,the diagonals bisect each o

Vedclass · Accepted Answer

$(4, 7, 6)$

Vedclass · Answer

(B) Let the vertices of the parallelogram be $A(1, 2, 3)$,$B(-1, -2, -1)$,$C(2, 3, 2)$,and $D(x, y, z)$.In a parallelogram,the diagonals bisect each other. This means the midpoint of diagonal $AC$ is the same as the midpoint of diagonal $BD$.Midpoint of $AC = (\frac{1+2}{2}, \frac{2+3}{2}, \frac{3+2}{2}) = (\frac{3}{2}, \frac{5}{2}, \frac{5}{2})$.Midpoint of $BD = (\frac{-1+x}{2}, \frac{-2+y}{2}, \frac{-1+z}{2})$.Equating the midpoints:$\frac{-1+x}{2} = \frac{3}{2} \implies -1+x = 3 \implies x = 4$.$\frac{-2+y}{2} = \frac{5}{2} \implies -2+y = 5 \implies y = 7$.$\frac{-1+z}{2} = \frac{5}{2} \implies -1+z = 5 \implies z = 6$.Thus,the fourth vertex $D$ is $(4, 7, 6)$.

If the three consecutive vertices of a parallelogram are $A(1, 2, 3)$,$B(-1, -2, -1)$,and $C(2, 3, 2)$,then its fourth vertex is:

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