Q100 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

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

  • A
    (1, 3, 3)
  • B
    $(1,3,-3)$
  • C
    $(3,3,-1)$
  • Correct answer
    D
    $(3,3,1)$

Solution

$A=(2,4,-1), B=(3,6,-1), C=(4,5,1)$ Let $D=(x, y, z)$ Since, diagonals of a parallelogram are bisect each other. $\Rightarrow$ Mid-point of $A C=$ Mid-point of $B D$ $$ \begin{aligned} & \left(\frac{2+4}{2}, \frac{4+5}{2}, \frac{-1+1}{2}\right)=\left(\frac{3+x}{2}, \frac{6+y}{2}, \frac{-1+z}{2}\right) \\ & \left(3, \frac{9}{2}, 0\right)=\left(\frac{3+x}{2}, \frac{6+y}{2}, \frac{-1+z}{2}\right) \\ & \therefore \frac{3+x}{2}=3 ; \frac{6+y}{2}=\frac{9}{2} ; \frac{-1+z}{2}=0 \\ & 3+x=6 ; 6+y=9 ;-1+z=0 \\ & x=3 ; y=3 ; z=1 \end{aligned} $$ $\therefore$ Fourth vertex $D=(3,3,1)$. Hence, option (d) is correct.

Hence, the correct answer is option: 'D'.

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