Question

Three vertices of a parallelogram ABCD are A$(3,-1,2),B(1,2,-4)andC(-1,1,2)$. Which of the follwing could be the fourth vertex ?

• A. $(1,-2,-8)$
• B. $(-1,-2,8)$
• C. $(1,-2,8)$
• D. $(1,2,8)$

Answer 1

The correct option is C

$(1,-2,8)$

Three vertices of parallelogram are given

Let fourth vertex be D$(x,y,z)$

we know that diagonals of parallelogram bisect each other.

O is the midpoint of the diagonal AC & BD

Let's calculate the coordinate of point O using diagonal AC.since,O is the midpoint of line AC

O$(x=\frac{3-1}{2},y=\frac{-1+1}{2},z=\frac{2+2}{2})$

O$(1,0,2)$

similarly we can also calculate the coordinate of point O using diagonal BD

O$(\frac{x+1}{2},\frac{y+2}{2},\frac{z-4}{2})$

These coordinates should be equal

$\frac{x+1}{2}=1,\frac{y+2}{2}=0,\frac{z-4}{8}=2$

$x=1,y=-2,z=8$

Coordinate of fourth vertex D is $(1,-2,8)$

We can form a parallelogram with AC as one side. In that case, the coordinates of D will be different. But since we got answer as one of the options, we stop here. Otherwise we would consider other possibilities