Q. 14.3( 39 Votes )

# Three vertices of

Answer :

Given: ABCD is a parallelogram, with vertices A (3, -1, 2), B (1, 2, -4), C (-1, 1, 2).

⇒ x_{1} = 3, y_{1} = -1, z_{1} = 2; x_{2} = 1, y_{2} = 2, z_{2} = -4; x_{3} = -1, y_{3} = 1, z_{3} = 2

Let the coordinates of the fourth vertex be D (x, y, z).

We know that the diagonals of a parallelogram bisect each other, so the mid points of AC and BD are equal, i.e. Midpoint of AC = Midpoint of BD ……….(i)

Now, by Midpoint Formula, we know that the coordinates of the mid-point of the line segment joining two points P (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) are .

So, we have

Coordinates of the midpoint of AC

Coordinates of the midpoint of BD

So, using (i), we have

⇒ 1 + x = 2, 2 + y = 0, -4 + z = 4

⇒ x = 1, y = -2, z = 8

Hence, the coordinates of the fourth vertex is D (1, -2, 8).

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