Question

Three vertices of a parallelogram ABCD are A ( 3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). The coordinates of fourth vertex D are ______.

Options

• (1, 1, 1)

• (1, –2, 8)

• (2, –2, 6)

• (1, 0, 2)

Answer 1

Three vertices of a parallelogram ABCD are A ( 3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). The coordinates of fourth vertex D are (1, –2, 8).

Explanation:

Let D (x, y, z) be the required point, Then, the mid-point of diagonal BD is

`((x + 1)/2, (y + 2)/2, (z - 4)/2)`

Also, the mid-point of diagonal AC is

`((3 - 1)/2, (-1 + 1)/2, (2 + 2)/2)` i.e., (1, 0, 2)

But, the mid-points of the diagonals of a parallelogram always coincide.

∴ `(x + 1)/2` = 1, `(y + 2)/2` = 0 and `(z - 4)/2` = 2

So, x = 1, y = –2 and z = 8.

Hence, the required point is D(1, –2, 8).