If the points (2,4,-1), (3,6,-1) and (4,5,1) | vedclass

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(A) Let the vertices of the parallelogram be $A(2,4,-1)$,$B(3,6,-1)$,and $C(4,5,1)$.Let the fourth vertex be $D(x, y, z)$.In a parallelogram,the diago

Vedclass · Accepted Answer

$(3,3,1)$

Vedclass · Answer

(A) Let the vertices of the parallelogram be $A(2,4,-1)$,$B(3,6,-1)$,and $C(4,5,1)$.Let the fourth vertex be $D(x, y, z)$.In a parallelogram,the diagonals bisect each other,which means the midpoint of diagonal $AC$ is the same as the midpoint of diagonal $BD$.Midpoint of $AC = \left( \frac{2+4}{2}, \frac{4+5}{2}, \frac{-1+1}{2} \right) = \left( 3, \frac{9}{2}, 0 \right)$.Midpoint of $BD = \left( \frac{3+x}{2}, \frac{6+y}{2}, \frac{-1+z}{2} \right)$.Equating the midpoints:$\frac{3+x}{2} = 3 \Rightarrow 3+x = 6 \Rightarrow x = 3$.$\frac{6+y}{2} = \frac{9}{2} \Rightarrow 6+y = 9 \Rightarrow y = 3$.$\frac{-1+z}{2} = 0 \Rightarrow -1+z = 0 \Rightarrow z = 1$.Thus,the fourth vertex $D$ is $(3,3,1)$.

If the points $(2,4,-1), (3,6,-1)$ and $(4,5,1)$ are three consecutive vertices of a parallelogram,then its fourth vertex is

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