Three consecutive vertices of a parallelogram ABCD are A (6, – 2, 4), B (2, 4, – 8), C (–2, 2, 4). Find the coordinates of the fourth vertex.

Home » Three consecutive vertices of a parallelogram ABCD are A (6, – 2, 4), B (2, 4, – 8), C (–2, 2, 4). Find the coordinates of the fourth vertex.

CBSE | Class 11 | Introduction to Three Dimensional Geometry | Maths | NCERT Exemplar

Three consecutive vertices of a parallelogram ABCD are A (6, – 2, 4), B (2, 4, – 8), C (–2, 2, 4). Find the coordinates of the fourth vertex.

Solution:

The three consecutive vertices of a parallelogram $A B C D$ are as given
$A(6,-2,4), B$ (2, $4,-8), C(-2,2,4)$
Suppose the forth vertex be $D(x, y, z)$
Midpoint of diagonal $A C=P\left(\frac{6-2}{2}, \frac{-2+2}{2}, \frac{4+4}{2}\right)=P(2,0,4)$
Midpoint of diagonal $B D=P\left(\frac{x+2}{2}, \frac{y+4}{2}, \frac{z-8}{2}\right)=P(2,0,4)$
$\begin{array}{l} \Rightarrow \frac{x+2}{2}=2 \Rightarrow x=2 \\ \Rightarrow \frac{y+4}{2}=0 \Rightarrow y=-4 \\ \Rightarrow \frac{z-8}{2}=4 \Rightarrow z=16 \end{array}$
$\therefore \mathrm{D}(2,-4,16)$ is the fourth vertex.